On Soft Near-Prime Int-Ideals and Soft 1-Absorbing Prime Int-Ideals With Applications

Abstract

In this study, we aimed to introduce two different generalizations of the soft prime int-ideal and clarify the relationships between the soft prime int-ideal and the substructures of a ring. First, we explored new algebraic features of the soft prime int-ideal. We defined and exemplified essential concepts such as soft nilpotent elements, soft idempotent elements, nonexplicit soft int-ideals, one-to-one soft sets, and soft zero divisors of a soft int-ideal. Then, we investigated the behavior of soft nilpotent and soft idempotent elements in relation to the soft prime int-ideal. We then defined a soft near-prime int-ideal, which is a generalization of the soft prime int-ideal through soft radicals, and examined its properties. Furthermore, we analyzed the relationships between the soft prime int-ideal and the soft near-prime int-ideal, showing that every soft prime int-ideal is a soft near-prime int-ideal, but the converse does not hold. In addition, we proposed the concept of a soft 1-absorbing prime int-ideal as another generalization of the soft prime int-ideal and studied its basic properties. We proved that the radicals of soft near-prime int-ideals and soft 1-absorbing prime int-ideals are soft prime int-ideals. Lastly, we examined soft homomorphic images and preimages of these new structures.

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1. Introduction

Real-life problems in many fields, such as medical sciences, logic, control engineering, and economics, include uncertainty. In recent years, scientists have recognized these problems and developed practical mathematical tools, such as fuzzy sets [1], rough sets [2], and soft sets [3], to struggle with these problems. Molodtsov [3] introduced soft sets, and Maji et al. described their operations in [4]. Ali et al. [5, 6] subsequently characterized new operations using soft sets. Since then, many authors have applied this unique and highly useful tool to various fields. Çağman and Enginoğlu [7] applied soft sets to a decision-making methods using this new concept. Following this work, many algebraic structures were defined, the most notable being the soft group, introduced by Aktaş and Çağman [8]. In addition, the soft ring was explored by Acar et al. [9]. Bera et al. [10] studied the soft congruence relation over a lattice. Later, a new perspective on the soft group was provided by Çağman et al.’s description of the soft int-group [11]. Ullah et al. [12] studied the soft uni-AG-group and investigated its properties. Karaaslan et al. [13] described the bipolar soft group and examined its features. Karaaslan et al. [14] introduced the normal bipolar soft subgroup and discussed its properties. In [15], soft int-ring/int-ideal was presented, and Çıtak and Çağman investigated their algebraic properties. Ghosh et al. [16] explored the soft prime int-ideal and the soft radical, addressing their basic properties. Ameen et al. [17] examined the relationship between ordinary and soft algebras. Al-Shami et al. [18] studied the connection between ordinary and soft $\sigma$-algebras. Many scientists have continued to implement their findings in this area [19–23]. Given the functionality of this concept, there have been numerous developments in topology. Çağman et al. [24] described soft topology and investigated its properties. Georgiou and Megaritis [25] introduced new notions in soft topology, such as soft closure, soft boundary, and soft homeomorphism, and examined their properties. Mondal et al. [26] studied Tietze’s extension theorem and Urysohn’s lemma in soft topology. Furthermore, Memiş et al. [27] proposed a new approach to fuzzy soft decision-making using FPFS-kNN. Dalkılıç and Demirtaş [28] examined the management of the decision-making process regarding the uncertainty problems that arose during the COVID-19 pandemic. In addition, they [29] established a relation between the inoculation commitment and COVID-19, supporting their findings with similar relational measures. Parmaksız et al. [30] investigated treatment and planning during COVID-19 using fuzzy parameterized fuzzy soft matrices.

Ideals are significant tools in ring theory because the more we understand about an ideal $I$ of a ring $R$, the more we understand about the structure of $R$. One of the most essential types of ideals is the prime ideal. Yassine et al. [31] developed 1-absorbing prime ideals and investigated their algebraic features. Koç et al. [32] analyzed some properties by defining a weakly 1-absorbing prime ideal. Groenewald [33, 34] studied relationships between 1-absorbing prime ideals/submodules and weakly 1-absorbing prime ideals/submodules in noncommutative rings. In recent years, the research on the 1-absorbing prime ideal has become increasingly prominent [35–38].

Algebraic structures are widely used and have a significant impact on many areas of mathematics, including theoretical physics, computer science, control engineering, information sciences, coding theory, topological spaces, and more. On the other hand, soft sets play an essential role in solving problems involving uncertainty. Many studies on soft sets have been conducted in areas such as decision-making, engineering, topology, algebra, and others. After the introduction of soft sets, they attracted significant attention from researchers studying algebraic structures. The behavior of soft sets in algebraic structures has become a topic of considerable curiosity and has begun to be actively investigated. Moreover, ideals are a fundamental tool in ring theory, and the prime ideal is particularly important. A review of the literature reveals that the subalgebraic structures of soft algebraic structures have not received sufficient attention, which represents a significant gap in the field. It is believed that many structures in this area remain to be explored.

The motivation for writing this paper is primarily to examine the behavior of the prime ideals, which hold an important place in the theory of rings on soft sets. Secondly, the paper aims to present a new framework for generating soft algebraic concepts, such as the soft radical and soft nilpotent elements, which were obtained in this study. In addition, the soft prime int-ideal is generalized to the soft near-prime int-ideal and the soft 1-absorbing prime int-ideal. Thus, the study is made unique by elevating the identity to the soft prime ideal. By exploring the subalgebraic structures of the ring $R$ through the specific properties of the new soft algebraic structures defined here, it is clearly demonstrated that the permeability between the soft and classical algebraic structures is very strong. Another motivation for this study is to develop new perspectives through generalizations of the soft prime int-ideal, which is crucial for rings, and to clarify the relationships between these new concepts and their classical counterparts. This study aims to contribute to the literature by obtaining two different generalizations of the soft prime int-ideal and exploring the relationships between the soft prime int-ideal and ring substructures. Unlike previous studies, the characteristic features of the ring $R$ are frequently used in this work, which is another unique aspect of our study. Throughout the paper, we aimed to demonstrate that the relationship between the ring $R$ and soft algebraic structures is close. These new soft ideal types and soft algebraic structures can be applied in future studies.

In Section 2, we included basic definitions and theorems. In Section 3, we introduced new concepts such as soft nilpotent elements, soft idempotent elements, soft zero divisors, nonexplicit soft int-ideals, and one-to-one soft sets. We also investigated their characterizations of soft prime int-ideals and their effects on ring $R$. In Section 4, we first defined and discussed the soft near-prime int-ideal, a generalization of the soft prime int-ideal, with the help of soft radicals. In addition, we demonstrated that a soft prime int-ideal is also a soft near-prime int-ideal. By providing a counterexample, we showed that a soft near-prime int-ideal does not have to be a soft prime int-ideal. We proved that if $R$ is a commutative ring, the ${\eta }_R$’s soft radical is a soft prime int-ideal, which is one of the most significant properties of the soft near-prime int-ideal ${\eta }_R$. In Section 5, we defined another generalization of the soft prime int-ideal, namely, the soft 1-absorbing prime int-ideal. We confirm that the soft prime int-ideal is also a soft 1-absorbing prime int-ideal. However, we showed using a counterexample that the converse is not valid. At the same time, we showed that if $R$ is a commutative ring, then the radical of a soft 1-absorbing prime int-ideal is a soft prime int-ideal. We also discussed the behavior of soft nilpotent elements, soft zero divisors, and one-to-one soft sets on this soft ideal. We studied that another important feature of a soft 1-absorbing prime int-ideal ${\eta }_R$ is that if there is a soft 1-absorbing prime int-ideal and $I$ is an ideal of $R$, then there exist a soft 1-absorbing prime int-ideal of $R/I$ with the same features as ${\eta }_R$.

2. Preliminaries

This section provides some basic notions that will be required in the following sections. Throughout this paper, let $M$ be a universal set, $C$ be a set of parameters, $Z\subseteq C$, and $\mathcal{P}M$ be the power set of $M$.

Definition 1.

(see [3]). Let ${\eta }_Z:C⟶\mathcal{P}M$ be a function. An order pair ${\eta }_Z=c,{\eta }_{Z}c:c\in C$ is called a soft set of $C$ where ${\eta }_{Z}c=\varnothing$ for $c\notin Z$.

Throughout this study, the set of all soft sets will be symbolized by S$M={\eta }_Z\mid {\eta }_Z:C⟶\mathcal{P}M$.

Definition 2.

(see [7]). Let ${\eta }_Z,{\eta }_W\in SM$. The soft set ${\eta }_Z$ is called a soft subset of ${\eta }_W$ if ${\eta }_{Z}c\subseteq {\eta }_{W}c$ for all $c\in C$, and it is symbolized by ${\eta }_Z\tilde{\subseteq }{\eta }_W$.

Definition 3.

(see [11]). Let $⊺:Z⟶W$ be a function and ${\eta }_Z,{\eta }_W\in SM$. The soft set $⊺{\eta }_Z$ is called a soft image of ${\eta }_Z$ under $⊺$ where$$\begin{array}{}\text{(1)}& ⊺{\eta }_{Z}w=\begin{array}{cc}\cup {\eta }_{Z}z:z\in Z,⊺z=w,& \text{if\hspace{1em}}{⊺}^{-1}w\ne \varnothing ,\\ \varnothing ,& \text{otherwise},\end{array}\end{array}$$for all $w\in W$. The soft set ${⊺}^{-1}{\eta }_W$ is called a soft preimage of ${\eta }_W$ under $⊺$ where ${⊺}^{-1}{\eta }_{W}z={\eta }_W⊺z$ for all $z\in Z$.

The definitions of soft int-group, soft int-ring, and soft int-ideal are described below.

Definition 4.

(see [11]). Let $W$ be a group and ${\eta }_W\in SM$. The soft set ${\eta }_W$ is called a soft intersection group (or shortly soft int-group) if ${\eta }_W{w}_i{w}_k\supseteq {\eta }_W{w}_i\cap {\eta }_W{w}_k$ and ${\eta }_W{w}_i^{-1}={\eta }_W{w}_i$ for all $w_i,w_k\in W$.

Definition 5.

(see [15]). Let $J,+,\cdot$ be a ring and ${\eta }_J\in SM$. The soft set ${\eta }_J$ is called a soft intersection ring (or shortly soft int-ring) if ${\eta }_W{w}_i+w_k\supseteq {\eta }_W{w}_i\cap {\eta }_W{w}_k$, ${\eta }_W-w_i={\eta }_W{w}_i$, and ${\eta }_W{w}_i{w}_k\supseteq {\eta }_W{w}_i\cap {\eta }_W{w}_k$ for all $w_i,w_k\in W$.

Definition 6.

(see [15]) Let ${\eta }_J$ be a soft int-ring. The soft int-ring ${\eta }_J$ is called a soft right int-ideal if ${\eta }_J{j}_i{j}_k\supseteq {\eta }_J{j}_k$ for all $j_i,j_k\in J$, and ${\eta }_J$ is called a soft left int-ideal if ${\eta }_J{j}_i{j}_k\supseteq {\eta }_J{j}_i$ for all $j_i,j_k\in J$. In addition, ${\eta }_J$ is called a soft int-ideal of $J$ if ${\eta }_J$ is both the soft right int-ideal and soft left int-ideal.

We proposed the following theorems for use in the main results.

Theorem 1 (see [15]).

Let $J$ be a ring and ${\eta }_J\in SM$. The soft set ${\eta }_J$ is a soft int-ideal iff

1. ${\eta }_J{j}_i-j_k\supseteq {\eta }_J{j}_i\cap {\eta }_J{j}_k$

2. ${\eta }_J{j}_i{j}_k\supseteq {\eta }_J{j}_i\cup {\eta }_J{j}_k$

for all $j_i,j_k\in J$.

Theorem 2 (see [15]).

Let ${\eta }_J$ be a soft int-ring/int-ideal. For all $j_i\in J$, ${\eta }_{J}0\supseteq {\eta }_R{j}_i$.

Theorem 3 (see [15]).

Let $⊺:J⟶T$ be onto-ring homomorphism and ${\eta }_J$ be a soft int-ideal. The soft sets $⊺{\eta }_J$ and ${⊺}^{-1}{\eta }_T$ are soft int-ideals.

Theorem 4 (see [15]).

Let $J$ be a unitary ring. For all $j_i\in J$, ${\eta }_J{j}_i\supseteq {\eta }_{J}1$ if ${\eta }_J$ is a soft int-ideal.

Theorem 5 (see [15]).

Let $J$ be a division ring. For all $0\ne j_i\in J$, ${\eta }_J{j}_i={\eta }_{J}1\subseteq {\eta }_{J}0$ if ${\eta }_J$ is a soft int-ideal.

The definition of an image is as follows:

Definition 7 (see [16]).

Let ${\eta }_Z\in SM$. An image of ${\eta }_Z$ is symbolized by $\text{Im}{\eta }_Z$ and described by $\text{Im}{\eta }_Z={\eta }_{Z}c:c\in C$.

Ghosh et al. [16] defined the soft int-ideal’s soft radical as follows.

Definition 8.

Let ${\eta }_J$ be a soft int-ideal such that $\text{Im}{\eta }_J={\eta }_{J_1},{\eta }_{J_2},\dots {\eta }_{J_n}$ and ${\eta }_{J_i}$ inclusion ideals of ${\eta }_J$ for $i=1,2,3,\dots n$. The soft set $\sqrt{{\eta }_J}$ is called a soft radical of ${\eta }_J$ such that$$\begin{array}{}\text{(2)}& \sqrt{{\eta }_J}{j}_i=\underset{j_i\in \sqrt{{\eta }_{J_i}}}{\cup }{\eta }_{J_i},\end{array}$$i.e., the union is taken over all those ${\eta }_{J_i}$ such that $j_i\in \sqrt{{\eta }_{J_i}}$ and $\sqrt{{\eta }_{J_1}}$, $\sqrt{{\eta }_{J_2}}$, $\dots$, $\sqrt{{\eta }_{J_n}}$ are the radicals of ${\eta }_{J_1},{\eta }_{J_2},\dots ,{\eta }_{J_n}$, respectively.

Theorem 6 (see [16]).

Let $J$ be a ring, ${\eta }_J$ be a soft int-ideal, and $\sqrt{{\eta }_J}$ be a soft radical of ${\eta }_J$. Then,$$\begin{array}{}\text{(3)}& \sqrt{{\eta }_J}{j}_i=\underset{n\mathbb{\in }\mathbb{N}}{\cup }{\eta }_J{j}_i^n,\end{array}$$for all $j_i\in J$.

The following basic properties of soft radicals will be useful in subsequent results.

Theorem 7 (see [16]).

Let $J$ be a ring, ${\eta }_J$ be a soft int-ideal, and $\sqrt{{\eta }_J}$ be a soft radical of ${\eta }_J$. Thus, the following hold:

1. $\sqrt{{\eta }_J}0={\eta }_{J}0$

2. $\sqrt{{\eta }_J}$ is a soft int-ideal and $\sqrt{{\eta }_J}\tilde{\supseteq }{\eta }_J$

3. $\sqrt{\sqrt{{\eta }_J}}=\sqrt{{\eta }_J}$

4. $\sqrt{{\eta }_J}{j}_i\supseteq {\eta }_J{j}_i^n$ for arbitrary $j_i\in J$ and for arbitrary $n\mathbb{\in }\mathbb{N}$

5. There exists some $n\mathbb{\in }\mathbb{N}$ where $\sqrt{{\eta }_J}{j}_i={\eta }_J{j}_i^n$ for arbitrary $j_i\in J$

Definition 9.

(see [16]). Let ${\eta }_J$ be a soft int-ideal. ${\eta }_J$ is called a soft prime int-ideal if either ${\eta }_J{j}_i{j}_k={\eta }_J{j}_i$ or ${\eta }_J{j}_i{j}_k={\eta }_J{j}_k$ for all $j_i,j_k\in J$.

Theorem 8 (see [16]).

Let ${\eta }_J$ be a soft int-ideal. Then, ${\eta }_J$ is a soft prime int-ideal iff ${\eta }_J{j}_i{j}_k={\eta }_J{j}_i\cup {\eta }_J{j}_k$ for all $j_i,j_k\in J$.

Definition 10 (see [31]).

Let $J$ be a unitary ring and $W$ be a proper ideal of $J$. $W$ is called a 1-absorbing prime ideal of $J$, if for all irreversible elements, $w_i,w_j,w_k\in J$ such that $w_i{w}_j{w}_k\in W$, then either $w_i{w}_j\in W$ or $w_k\in W$.

Theorem 9 (see [31]).

Let $J$ be a unitary and commutative ring, $W$ be a 1-absorbing prime ideal of $J$, and $w_i\in J$. Then, $w_i^n\in W⟹w_i^2\in W$ for all $n\ge 3\in \mathbb{N}$ and $w_j\in \sqrt{W}\setminus W⟹w_j^2\in W$.

3. Soft Prime Int-Ideals

This section investigates some algebraic properties of the soft prime int-ideal and explores its features. Throughout this section, let $R$ be a ring. In addition, the zero of $R$ is denoted by 0, and the unit of $R$ is denoted by 1. Lastly, the suffix “int” will no longer be used in this section.

Definition 11.

Let ${\eta }_R$ be a soft set and $\mathfrak{I}\mathfrak{\subseteq }M$. Then, ${\eta }_R$ is a soft ideal if ${\eta }_R{r}_i=\mathfrak{I}$ for all $r_i\in R$. These kinds of soft ideals are also soft prime ideals. Such ideals are called explicit soft ideals. Otherwise, all these kinds of ideals are called nonexplicit soft ideals.

Example 1.

Let $R={\mathbb{Z}}_4$ and $M=a,b,c,d$. Let the fuction ${\eta }_R:R⟶PM$ be defined such that ${\eta }_R\overline{0}={\eta }_R\overline{1}={\eta }_R\overline{2}={\eta }_R\overline{3}=a,c,d$. ${\eta }_R$ is an explicit soft ideal because$$\begin{array}{}\text{(4)}& {\eta }_R\overline{r_i}-\overline{r_j}\supseteq {\eta }_R\overline{r_i}\cap {\eta }_R\overline{r_j},\end{array}$$and$$\begin{array}{}\text{(5)}& {\eta }_R\overline{r_i}\overline{r_j}={\eta }_R\overline{r_i}\cup {\eta }_R\overline{r_j},\end{array}$$for all $\overline{r_i},\overline{r_j}\in R$. Eventually, ${\eta }_R$ is a soft prime ideal.

Definition 12.

Let ${\eta }_R$ be a nonexplicit soft ideal. $r_i,{\eta }_R{r}_i$ is called the ${\eta }_R$’s soft nilpotent element, if there is an $n\in {\mathbb{Z}}^{+}$ such that$$\begin{array}{}\text{(6)}& {\eta }_R{r}_i\cup {\eta }_R{r}_i^2,\dots ,{\eta }_R{r}_i^n=\underset{j=1}{\overset{n}{\cup }}{\eta }_R{r}_i^j={\eta }_{R}0,\end{array}$$and it is indicated by $r_i,{\eta }_R{r}_i\tilde{\in }\text{Nil}{\eta }_R$. In other words,$$\begin{array}{}\text{(7)}& \text{Nil}{\eta }_R=r_i,{\eta }_R{r}_i\tilde{\in }{\eta }_R:\sqrt{{\eta }_R}{r}_i={\eta }_{R}0.\end{array}$$

Example 2.

Let $R=r_i,r_j,r_k,r_t$ and the binary operations of “$+$” and “$\cdot$” described on $R$ be as follows:

Obviously, $R$ is a ring. Let the function ${\eta }_R:R⟶P{S}_3$ be defined as follows:$$\begin{array}{}\text{(8)}& {\eta }_R{r}_i=S_3,{\eta }_R{r}_j=S_3\setminus 1,{\eta }_R{r}_k={\eta }_R{r}_t=S_3\setminus 1,12,13,\end{array}$$${\eta }_R$ is a nonexplicit soft ideal because

• ${\eta }_R{r}_j+r_k={\eta }_R{r}_t\supseteq {\eta }_R{r}_j\cap {\eta }_R{r}_k$

• ${\eta }_R{r}_j+r_t={\eta }_R{r}_k\supseteq {\eta }_R{r}_j\cap {\eta }_R{r}_t$

• ${\eta }_R{r}_k+r_t={\eta }_R{r}_j\supseteq {\eta }_R{r}_k\cap {\eta }_R{r}_t$

• ${\eta }_R{r}_j+r_j={\eta }_R{r}_i\supseteq {\eta }_R{r}_j$

• ${\eta }_R{r}_k+r_k={\eta }_R{r}_i\supseteq {\eta }_R{r}_k$

• ${\eta }_R{r}_t+r_t={\eta }_R{r}_i\supseteq {\eta }_R{r}_t$

• ${\eta }_R{r}_j={\eta }_R-r_j$

• ${\eta }_R{r}_k={\eta }_R-r_k$

• ${\eta }_R{r}_t={\eta }_R-r_t$

• ${\eta }_R{r}_k{r}_t={\eta }_R{r}_t{r}_k={\eta }_R{r}_k\supseteq {\eta }_R{r}_k\cup {\eta }_R{r}_t$

Similarly, we can see that other conditions are satisfied. Thus, ${\eta }_R$ is a nonexplicit soft ideal. Now, we determine the soft nilpotent element of ${\eta }_R$:

• $\sqrt{{\eta }_R}{r}_i={\cup }_{m\mathbb{\in }\mathbb{N}}{\eta }_R{r_i}^m={\eta }_R{r}_i⟹r_i,{\eta }_R{r}_i\tilde{\in }\text{Nil}{\eta }_R$

• $\sqrt{{\eta }_R}{r}_j={\cup }_{m\mathbb{\in }\mathbb{N}}{\eta }_R{r_j}^m={\eta }_R{r}_j\cup {\eta }_R{r_j}^2\cup ,\dots ,{\eta }_R{r_j}^n={\eta }_R{r}_i⟹r_j,{\eta }_R{r}_j\tilde{\in }\text{Nil}{\eta }_R$

• $\sqrt{{\eta }_R}{r}_k={\cup }_{m\mathbb{\in }\mathbb{N}}{\eta }_R{r_k}^m={\eta }_R{r}_k\cup {\eta }_R{r_k}^2\cup ,\dots ,{\eta }_R{r_k}^n={\eta }_R{r}_k\ne {\eta }_R{r}_i⟹r_k,{\eta }_R{r}_k\tilde{\notin }\text{Nil}{\eta }_R$

• $\sqrt{{\eta }_R}{r}_t={\cup }_{m\mathbb{\in }\mathbb{N}}{\eta }_R{r_t}^m={\eta }_R{r}_t\cup {\eta }_R{r_t}^2\cup ,\dots ,{\eta }_R{r_t}^n={\eta }_R{r}_t\ne {\eta }_R{r}_i⟹r_t,{\eta }_R{r}_t\tilde{\notin }\text{Nil}{\eta }_R$

In conclusion,$$\begin{array}{}\text{(9)}& \text{Nil}{\eta }_R=r_i,{\eta }_R{r}_i,r_j,{\eta }_R{r}_j.\end{array}$$

Theorem 10.

Let ${\eta }_R$ be a nonexplicit soft prime ideal. If $r_i\in \text{Nil}R$, then $r_i,{\eta }_R{r}_i\tilde{\in }\text{Nil}{\eta }_R$.

Proof 1.

Let $r_i\in \text{Nil}R$. In this case, there is an $n\in {\mathbb{Z}}^{+}$ such that $r_i^n=0$. The following is obtained:$$\begin{array}{}\text{(10)}& {\eta }_R{r}_i^n={\eta }_{R}0⟹{\eta }_R{r}_i={\eta }_{R}0⟹\sqrt{{\eta }_R}{r}_i=\sqrt{{\eta }_R}0=\sqrt{{\eta }_R}{r}_i={\eta }_{R}0⟹r_i,{\eta }_R{r}_i\tilde{\in }\text{Nil}{\eta }_R.\end{array}$$

The converse of this theorem is not always valid. To prove this, we give a counterexample.

Example 3.

Let $M=S_3,R={\mathbb{Z}}_6$ and ${\eta }_R\overline{0}={\eta }_R\overline{2}={\eta }_R\overline{4}=S_3,{\eta }_R\overline{1}={\eta }_R\overline{3}={\eta }_R\overline{5}=1$. Thus, ${\eta }_R$ is a nonexplicit soft prime ideal and $\sqrt{{\eta }_R}\overline{2}={\eta }_R\overline{0}⟹\overline{2},{\eta }_R\overline{2}\tilde{\in }\text{Nil}{\eta }_R$, but $\overline{2}\notin \text{Nil}{\mathbb{Z}}_6$.

Definition 13.

Let ${\eta }_R$ be a nonexplicit soft ideal. $r_i,{\eta }_R{r}_i$ is called a soft idempotent element of ${\eta }_R$ if ${\eta }_R{r}_i^2={\eta }_R{r}_i$, and it is shown that $r_i,{\eta }_R{r}_i\tilde{\in }\text{Ide}{\eta }_R$. In other words,$$\begin{array}{}\text{(11)}& \text{Ide}{\eta }_R=r_i,{\eta }_R{r}_i\tilde{\in }{\eta }_R:{\eta }_R{r}_i^2={\eta }_R{r}_i.\end{array}$$

Example 4.

Let $R={\mathbb{Z}}_4$ and $M=S_3$. Let ${\eta }_R:R⟶P{S}_3$ be defined as follows:$$\begin{array}{}\text{(12)}& {\eta }_R\overline{0}=S_3,{\eta }_R\overline{2}=S_3\setminus 1,{\eta }_R\overline{1}={\eta }_R\overline{3}=S_3\setminus 1,123.\end{array}$$

Clearly, ${\eta }_R$ is a nonexplicit soft ideal. Now, we determine soft idempotent elements of ${\eta }_R$:

• ${\eta }_R{\overline{0}}^2={\eta }_R\overline{0}=S_3⟹\overline{0},{\eta }_R\overline{0}$ is a soft idempotent element of ${\eta }_R$

• ${\eta }_R{\overline{1}}^2={\eta }_R\overline{1}=S_3\setminus 1,123⟹\overline{1},{\eta }_R\overline{1}$ is a soft idempotent element of ${\eta }_R$

• ${\eta }_R{\overline{2}}^2={\eta }_R\overline{0}=S_3\ne {\eta }_R\overline{2}⟹\overline{2},{\eta }_R\overline{2}$ is not a soft idempotent element of ${\eta }_R$

• ${\eta }_R{\overline{3}}^2={\eta }_R\overline{1}=S_3\setminus 1,123={\eta }_R\overline{3}⟹\overline{3},{\eta }_R\overline{3}$ is a soft idempotent element of ${\eta }_R$

Theorem 11.

Let ${\eta }_R$ be a nonexplicit soft prime ideal. Then, ${\eta }_R=\text{Ide}{\eta }_R$.

Proof 2.

By Theorem 8, ${\eta }_R{r}_i^n={\eta }_R{r}_i$ for all $r_i\in R$. It is easy to see that this equation is valid for $n=2$. Thus, ${\eta }_R=$ Ide ${\eta }_R$.

Definition 14.

Let ${\eta }_R$ be a soft ideal and ${\eta }_R{r}_i\ne {\eta }_{R}0$. $r_i,{\eta }_R{r}_i$ and $r_i,{\eta }_R{r}_j$ are called soft zero divisors of ${\eta }_R$ if there is an $r_j\in R$ such that ${\eta }_R{r}_j\ne {\eta }_{R}0$ and ${\eta }_R{r}_i\cup {\eta }_R{r}_j={\eta }_{R}0$. A soft set of all soft zero divisors of ${\eta }_R$ are shown as $\text{ZD}{\eta }_R$.

Example 5.

Let $R=r_i,r_j,r_k,r_t$ and the binary operations of “+” and “$\cdot$” described on $R$ be as follows:

It is obvious that $R,+,\cdot$ is a ring. Let ${\eta }_R:R⟶P{S}_3$ function be defined as follows:$$\begin{array}{}\text{(13)}& {\eta }_R{r}_i=S_3,{\eta }_R{r}_j=1,12,23,132,{\eta }_R{r}_k=1,{\eta }_R{r}_t=1,13,123.\end{array}$$

It is easy to observe that ${\eta }_R$ is a soft ideal. Furthermore, since ${\eta }_R{r}_j\cup {\eta }_R{r}_t={\eta }_R{r}_i$, then $r_j,{\eta }_R{r}_j$ and $r_t,{\eta }_R{r}_t$ are soft zero divisors of ${\eta }_R$.

Definition 15.

Let ${\eta }_Z\in SM$. ${\eta }_Z$ is called a one-to-one soft set, if ${\eta }_Z{c}_1={\eta }_Z{c}_2$ implies $c_1=c_2$ for $c_1,c_2\in C$.

Example 6.

Let $R=r_i,r_j,r_k,r_t$ and the binary operations of “+” and “⋅” described on $R$ be as follows:

It is observed that $R,+,\cdot$ is a ring. Let ${\eta }_R:R⟶P{S}_3$ be defined as follows:$$\begin{array}{}\text{(14)}& {\eta }_R{r}_i=S_3,{\eta }_R{r}_j=1,12,{\eta }_R{r}_k=13,123,{\eta }_R{r}_t=23,132.\end{array}$$

It is obvious that ${\eta }_R$ is a one-to-one soft ideal.

Theorem 12.

Let ${\eta }_R$ be a one-to-one soft prime ideal. If $r_i$ is a zero divisor of $R$, then $r_i,{\eta }_R{r}_i\tilde{\in }ZD{\eta }_R$.

Proof 3.

Let $r_i$ be a zero divisor of ring $R$. In that case, there is $r_j\in R\setminus 0$ such that ${\eta }_R{r}_j\ne {\eta }_{R}0$ and $r_i{r}_j=0$. In this situation,$$\begin{array}{}\text{(15)}& {\eta }_{R}0={\eta }_R{r}_i{r}_j={\eta }_R{r}_i\cup {\eta }_R{r}_j⟹r_i,{\eta }_R{r}_i\tilde{\in }ZD{\eta }_R.\end{array}$$

Theorem 13.

Let ${\eta }_R$ be a one-to-one soft prime ideal. Then, Nil $R=0$.

Proof 4.

Let $r_i\in$ Nil $R$. In this situation, there is $n\mathbb{\in }\mathbb{Z}$ such that $r_i^n=0$. Since ${\eta }_R$ is a one-to-one soft prime ideal,$$\begin{array}{}\text{(16)}& {\eta }_{R}0={\eta }_R{r}_i^n={\eta }_R{r}_i⟹r_i=0.\end{array}$$

Theorem 14.

Let ${\eta }_R$ be a one-to-one soft prime ideal. Then, Nil ${\eta }_R=0,{\eta }_{R}0$.

Proof 5.

Let $r_j,{\eta }_R{r}_j\in \text{Nil}{\eta }_R$. Then, $\sqrt{{\eta }_R}{r}_j={\eta }_{R}0$. Since ${\eta }_R$ is a one-to-one soft prime ideal,$$\begin{array}{}\text{(17)}& {\eta }_{R}0=\sqrt{{\eta }_R}{r}_j={\eta }_R{r}_j⟹r_j=0⟹\text{Nil}{\eta }_R=0,{\eta }_{R}0.\end{array}$$

Theorem 15.

Let ${\eta }_R$ be a soft prime ideal, and$$\begin{array}{}\text{(18)}& K=r_i\in R:r_j\in R\text{\hspace{0.17em}where\hspace{0.17em}}{\eta }_R{r}_i\cup {\eta }_R{r}_j={\eta }_{R}0.\end{array}$$

Then, $K$ is a prime ideal of $R$.

Proof 6.

It is easy to observe that $0\in K$ and $K\ne \varnothing$. Let $r_i,r_j\in K$. There are $s_i,s_j\in R$ such that ${\eta }_R{r}_i\cup {\eta }_R{s}_i={\eta }_{R}0$ and ${\eta }_R{r}_j\cup {\eta }_R{s}_j={\eta }_{R}0$. Under these situations,$$\begin{array}{c}\text{(19)}& {\eta }_R{r}_i-r_j{s}_i{s}_j={\eta }_R{r}_i{s}_i{s}_j-r_j{s}_j{s}_i\supseteq {\eta }_R{r}_i{s}_i{s}_j\cap {\eta }_R{r}_j{s}_j{s}_i\\ =\underset{={\eta }_{R}0}{\underbrace{{\eta }_R{r}_i\cup {\eta }_R{s}_i}}\cup {\eta }_R{s}_j\cap \underset{={\eta }_{R}0}{\underbrace{{\eta }_R{r}_j\cup {\eta }_R{s}_j}}\cup {\eta }_R{s}_i\\ ={\eta }_{R}0.\end{array}$$

Since ${\eta }_R{r}_i-r_j{s}_i{s}_j={\eta }_R{r}_i-r_j\cup {\eta }_R{s}_i{s}_j={\eta }_{R}0$, then $r_i-r_j\in K$.

Let $r\in R$. Then, ${\eta }_R{r}_{i}r{s}_i=\underset{={\eta }_{R}0}{\underbrace{{\eta }_R{r}_i\cup {\eta }_R{s}_i}}\cup {\eta }_{R}r={\eta }_{R}0⟹r_{i}r\in K$. Similarly, it is shown that $r{r}_i\in K$.

Let $r_n,r_m\in R$ such that $r_n{r}_m\in K$. Hence, there is $r_k\in R$ such that ${\eta }_R{r}_n{r}_m\cup {\eta }_R{r}_k={\eta }_{R}0$. Let $r_m\notin K$.$$\begin{array}{c}\text{(20)}& {\eta }_{R}0={\eta }_R{r}_n{r}_m\cup {\eta }_R{r}_k={\eta }_R{r}_n\cup {\eta }_R{r}_m\cup {\eta }_R{r}_k\\ ={\eta }_R{r}_n\cup {\eta }_R{r}_m{r}_k.\end{array}$$

Since ${\eta }_R{r}_n\cup {\eta }_R{r}_m{r}_k={\eta }_{R}0$, then $r_n\in K$.

Similarly, it is shown that if $r_n{r}_m\in K$ and $r_n\notin K$, then $r_m\in K$. Eventually, $K$ is a prime ideal of $R$.

Proposition 1.

Let ${\eta }_R$ be a soft prime ideal and $S=r_i\in R\mid {\eta }_R{r}_i=\mho$ be a subgroup of $R$. Then,

1. $S$ is an ideal of $R$, if ${\eta }_R{r}_k\subset \mho$ for all $r_k\in R\setminus S$

2. $S$ is a prime ideal of $R$, if $\cup {\eta }_R{r}_k\subset \mho$ for all $r_k\in R\setminus S$

Proof 7.

For all $r_i,r_j\in S$ and $r_k\in R\setminus S$,

 

• ${\eta }_R{r}_i{r}_j={\eta }_R{r}_i\cup {\eta }_R{r}_j=\mho \cup \mho =\mho ⟹r_i{r}_j\in S$

• ${\eta }_R{r}_i{r}_k={\eta }_R{r}_i\cup {\eta }_R{r}_k=\mho ⟹r_i{r}_k\in S$ and similarly, ${\eta }_R{r}_k{r}_i={\eta }_R{r}_k\cup {\eta }_R{r}_i=\mho ⟹r_k{r}_i\in S$

  As a result, $S$ is an ideal of $R$.

 

• ${\eta }_R{r}_i{r}_j={\eta }_R{r}_i\cup {\eta }_R{r}_j=\mho \cup \mho =\mho ⟹r_i{r}_j\in S$

• ${\eta }_R{r}_i{r}_k={\eta }_R{r}_i\cup {\eta }_R{r}_k=\mho ⟹r_i{r}_k\in S$ and similarly, $r_k{r}_i\in S$.

• Let $r_n,r_m\in R$ such that $r_n{r}_m\in S$ and $r_m\notin S$. Since $r_n{r}_m\in S$, $\mho ={\eta }_R{r}_n{r}_m={\eta }_R{r}_n\cup {\eta }_R{r}_m$. Because of $r_m\notin S$ and $\cup {\eta }_R{r}_k\subset \mho$, $r_n$ has to be an element of $S$.

Consequently, $S$ is a prime ideal of $R$.

Proposition 2.

Let ${\eta }_R$ be a soft ideal and $r_i,{\eta }_R{r}_i\tilde{\in }ZD{\eta }_R$. Then, $r_i{r}_j,{\eta }_R{r}_i{r}_j\tilde{\in }\text{Nil}{\eta }_R$ for $r_j\in R\setminus 0$.

Proof 8.

Since $r_i,{\eta }_R{r}_j\tilde{\in }ZD{\eta }_R$, there is an $r_j\in R\setminus 0$ such that ${\eta }_R{r}_j\ne {\eta }_{R}0$ and ${\eta }_R{r}_i\cup {\eta }_R{r}_j={\eta }_{R}0$. From here, it is observed that$$\begin{array}{c}\text{(21)}& {\eta }_R{r}_i\cup {\eta }_R{r}_j={\eta }_{R}0,\sqrt{{\eta }_R}{r}_i\cup \sqrt{{\eta }_R}{r}_j={\eta }_{R}0,\\ \sqrt{{\eta }_R}{r}_i{r}_j={\eta }_{R}0,\\ r_i{r}_j,{\eta }_R{r}_i{r}_j\tilde{\in }\text{Nil}{\eta }_R.\end{array}$$

4. Soft Near-Prime Int-Ideals and Its Some Algebraic Properties

This section investigates a generalization of soft prime int-ideals, which we call soft near-prime int-ideals. Lastly, the suffix “int” will no longer be used in this section.

Definition 16.

Let $R$ be a nonunitary ring and ${\eta }_R$ be a soft ideal. ${\eta }_R$ is called a soft near-prime ideal if ${\eta }_R{r}_i{r}_j=\sqrt{{\eta }_R}{r}_i\cup \sqrt{{\eta }_R}{r}_j$ for all $r_i,r_j\in R$.

Example 7.

Let $R=r_i,r_j,r_k$ and the binary operations “+” and “⋅” be defined as follows:

Based on these binary operations, $R$ is a nonunitary ring.

Let $M=S_3$ and ${\eta }_R=r_i,S_3,r_j,1,12,r_k,1,12$. It is easy to see that ${\eta }_R$ is a soft ideal. Furthermore, $\sqrt{{\eta }_R}{r}_j={\eta }_R{r}_j^2={\eta }_R{r}_i=S_3$ and $\sqrt{{\eta }_R}{r}_k={\eta }_R{r_k}^2={\eta }_R{r}_i=S_3$. Since $r_j{r}_k=r_k{r}_j=r_i$ and ${\eta }_R{r}_j{r}_k=S_3$. For this reason, we obtain that ${\eta }_R{r}_j{r}_k=\sqrt{{\eta }_R}{r}_j\cup \sqrt{{\eta }_R}{r}_k=S_3$. Hence, ${\eta }_R$ is a soft near-prime ideal.

Theorem 16.

Let ${\eta }_R$ be a soft near-prime ideal. Then, the following equalities are satisfied:$$\begin{array}{}\text{(22)}& {\eta }_R{r}_i^n=\sqrt{{\eta }_R}{r}_i=\sqrt{{\eta }_R}{r}_i^n,\end{array}$$for all $n\ge 2\in \mathbb{N}$.

Proof 9.

Since ${\eta }_R{r}_i{r}_j=\sqrt{{\eta }_R}{r}_i\cup \sqrt{{\eta }_R}{r}_j$ for all $r_i,r_j\in R$, we obtain$$\begin{array}{c}\text{(23)}& {\eta }_R{r}_i^2={\eta }_R{r}_i\cdot r_i=\sqrt{{\eta }_R}{r}_i\cup \sqrt{{\eta }_R}{r}_i=\sqrt{{\eta }_R}{r}_i,\\ \sqrt{{\eta }_R}{r}_i^2=\sqrt{\sqrt{{\eta }_R}}{r}_i\\ =\sqrt{{\eta }_R}{r}_i,\end{array}$$and$$\begin{array}{}\text{(24)}& {\eta }_R{r}_i^3={\eta }_R{r}_i^2\cdot r_i=\underset{=\sqrt{{\eta }_R}{r}_i}{\underbrace{\sqrt{{\eta }_R}{r}_i^2}}\cup \sqrt{{\eta }_R}{r}_i=\sqrt{{\eta }_R}{r}_i.\end{array}$$

Finally, it is easy to see that for all $n\ge 4\in \mathbb{Z}$,$$\begin{array}{}\text{(25)}& {\eta }_R{r}_i^n={\eta }_R{r}_i^{n-1}\cdot r_i=\underset{=\sqrt{{\eta }_R}{r}_i}{\underbrace{\sqrt{{\eta }_R}{r}_i^{n-1}}}\cup \sqrt{{\eta }_R}{r}_i=\sqrt{{\eta }_R}{r}_i.\end{array}$$

Moreover, we obtain ${\eta }_R{r}_i^n=\sqrt{{\eta }_R}{r}_i⟹\sqrt{{\eta }_R}{r}_i^n=\sqrt{\sqrt{{\eta }_R}}{r}_i=\sqrt{{\eta }_R}{r}_i$. Hence, the proof is completed.

Theorem 17.

Every soft prime ideal is a soft near-prime ideal.

Proof 10.

Let ${\eta }_R$ be a soft prime ideal. For all $r_i,r_j\in R$, ${\eta }_R{r}_i{r}_j={\eta }_R{r}_i\cup {\eta }_R{r}_j$ and ${\eta }_R{r}_i=\sqrt{{\eta }_R}{r}_i$. Hence,$$\begin{array}{}\text{(26)}& {\eta }_R{r}_i{r}_j={\eta }_R{r}_i\cup {\eta }_R{r}_j=\sqrt{{\eta }_R}{r}_i\cup \sqrt{{\eta }_R}{r}_j.\end{array}$$

Then, ${\eta }_R$ is a soft near-prime ideal.

The converse of this theorem is not always valid. To prove this, we gave a counterexample.

Example 8.

Consider the soft near-prime ideal ${\eta }_R$ in Example 7. ${\eta }_R$ is a soft near-prime ideal, but ${\eta }_R$ is not a soft prime ideal because$$\begin{array}{}\text{(27)}& {\eta }_R{r}_j{r}_k={\eta }_R{r}_i=S_3\ne {\eta }_R{r}_j\cup {\eta }_R{r}_k=1,12.\end{array}$$

Theorem 18.

Let $R$ be a nonunitary and commutative ring and ${\eta }_R$ be a soft near-prime ideal. The soft radical of ${\eta }_R$ is a soft prime ideal.

Proof 11.

From Theorem 16, we have ${\eta }_R{r}_i^2=\sqrt{{\eta }_R}{r}_i=\sqrt{{\eta }_R}{r}_i^2$ for all $r_i\in R$. Then, it is easy to see that$$\begin{array}{}\text{(28)}& \sqrt{{\eta }_R}{r}_i{r}_j={\eta }_R{r_i{r}_j}^2={\eta }_R{r}_i^2{r}_j^2=\sqrt{{\eta }_R}{r}_i^2\cup \sqrt{{\eta }_R}{r}_j^2=\sqrt{{\eta }_R}{r}_i\cup \sqrt{{\eta }_R}{r}_j.\end{array}$$

Therefore, $\sqrt{{\eta }_R}$ is a soft prime ideal.

Theorem 19.

Let $R$ and $S$ be nonunitary rings, $⊺:R⟶S$ be an onto-ring homomorphism, and ${\eta }_R$ be a soft near-prime ideal. Then, $⊺{\eta }_R$ is a soft near-prime ideal.

Proof 12.

By Theorem 3, we know that $⊺{\eta }_R$ is a soft ideal. Therefore, we only prove the near-primality of $⊺{\eta }_R$. For all $s_i,s_j\in S$,$$\begin{array}{c}\text{(29)}& ⊺{\eta }_R{s}_i{s}_j=\cup {\eta }_{R}r:r\in R,⊺r=s_i{s}_j=\cup {\eta }_R{r}_i{r}_j:r_i,r_j\in R,⊺r_i{r}_j=s_i{s}_j\\ =\cup {\eta }_R{r}_i{r}_j:r_i,r_j\in R,⊺r_i=s_i,⊺r_j=s_j\\ =\cup \sqrt{{\eta }_R}{r}_i\cup \sqrt{{\eta }_R}{r}_j:r_i,r_j\in R,⊺r_i=s_i,⊺r_j=s_j\\ =\cup \sqrt{{\eta }_R}{r}_i:r_i\in R,⊺r_i=s_i\cup \cup \sqrt{{\eta }_R}{r}_j:r_j\in R,⊺r_j=s_j\\ =\sqrt{⊺{\eta }_R}{s}_i\cup \sqrt{⊺{\eta }_R}{s}_j.\end{array}$$

Hence, $⊺{\eta }_R$ is a soft near-prime ideal.

Theorem 20.

Let $R$ and $S$ be nonunitary rings, $⊺:R⟶S$ be a ring homomorphism, and ${\eta }_S$ be a soft near-prime ideal. Then, ${⊺}^{-1}{\eta }_S$ is a soft near-prime ideal.

Proof 13.

By Theorem 3, we know that ${⊺}^{-1}{\eta }_S$ is a soft ideal. Therefore, we only proved the near-primality of ${⊺}^{-1}{\eta }_S$. For all $r_i,r_j\in R$,$$\begin{array}{c}\text{(30)}& {⊺}^{-1}{\eta }_S{r}_i{r}_j={\eta }_S⊺r_i{r}_j={\eta }_S⊺r_i⊺r_j\\ =\sqrt{{\eta }_S}⊺r_i\cup \sqrt{{\eta }_S}⊺r_j\\ =\sqrt{{⊺}^{-1}{\eta }_S}{r}_i\cup \sqrt{{⊺}^{-1}{\eta }_S}{r}_j.\end{array}$$

The proof is completed.

5. Soft 1-Absorbing Prime Int-Ideal and Its Some Algebraic Features

This section explores another generalization of soft prime int-ideals, which we call soft 1-absorbing prime int-ideals. Lastly, the suffix “int” will no longer be used in this section.

Definition 17.

Let $R$ be a nonunitary ring and ${\eta }_R$ be a soft ideal. ${\eta }_R$ is called a soft 1-absorbing prime ideal, if ${\eta }_R{r}_i{r}_j{r}_k={\eta }_R{r}_i{r}_j\cup {\eta }_R{r}_k$ for all $r_i,r_j,r_k\in R$.

Theorem 21.

Let ${\eta }_R$ be a soft 1-absorbing prime ideal. The following equalities are satisfied:$$\begin{array}{}\text{(31)}& {\eta }_R{r}_i^n={\eta }_R{r}_i^2=\sqrt{{\eta }_R}{r}_i,\end{array}$$for all $r_i\in R$ and $n\ge 2\in \mathbb{N}$.

Proof 14.

We use the inductive method. Since ${\eta }_R$ be a soft 1-absorbing prime ideal, we obtain ${\eta }_R{r}_i^3={\eta }_R{r}_i{r}_i{r}_i={\eta }_R{r}_i^2\cup {\eta }_R{r}_i={\eta }_R{r}_i^2$ for $n=3$. Assume that ${\eta }_R{r}_i^n={\eta }_R{r}_i^2$ for all $n>3$. In this situation,$$\begin{array}{}\text{(32)}& {\eta }_R{r}_i^{n+1}={\eta }_R{r}_i^{n-1}{r}_i{r}_i={\eta }_R{r}_i^n\cup {\eta }_R{r}_i={\eta }_R{r}_i^n={\eta }_R{r}_i^2,\end{array}$$and$$\begin{array}{}\text{(33)}& \sqrt{{\eta }_R}{r}_i=\underset{j=1}{\overset{n}{\cup }}{\eta }_R{r}_i^j={\eta }_R{r}_i\cup {\eta }_R{r}_i^2\cup \cdots \cup {\eta }_R{r}_i^n={\eta }_R{r}_i\cup {\eta }_R{r}_i^2={\eta }_R{r}_i^2.\end{array}$$

Theorem 22.

Let $R$ be a nonunitary and commutative ring and ${\eta }_R$ be a soft 1-absorbing prime ideal. $\sqrt{{\eta }_R}$ is a soft prime ideal.

Proof 15.

From the previous theorem, it is obtained that$$\begin{array}{}\text{(34)}& \sqrt{{\eta }_R}{r}_i{r}_j={\eta }_R{r_i{r}_j}^2={\eta }_R{r}_i^2{r}_j^2={\eta }_R{r}_i{r}_i{r}_j^2={\eta }_R{r}_i^2\cup {\eta }_R{r}_j^2=\sqrt{{\eta }_R}{r}_i\cup \sqrt{{\eta }_R}{r}_j,\end{array}$$for all $r_i,r_j\in R$.

Theorem 23.

Every soft prime ideal is a soft 1-absorbing prime ideal.

Proof 16.

Let $R$ be a nonunitary ring and ${\eta }_R$ be a soft prime ideal. Then, ${\eta }_R{r}_i{r}_j={\eta }_R{r}_i\cup {\eta }_R{r}_j$ for all $r_i,r_j\in R$. Since ${\eta }_R{r}_i{r}_j{r}_k={\eta }_R{r}_i{r}_j\cup {\eta }_R{r}_k$ for all $r_i,r_j,r_k\in R$, then ${\eta }_R$ is a soft 1-absorbing prime ideal.

The converse of this theorem is not always valid. To prove this, we give a counterexample.

Example 9.

Let $R=r_i,r_j,r_k,r_t$ and the binary operations “+” and “⋅” over $R$ be defined as in the tables below.

From these tables, $R$ is a nonunitary ring. Define the function ${\eta }_R:R⟶P{S}_3$ as follows:$$\begin{array}{}\text{(35)}& {\eta }_R{r}_i=S_3,{\eta }_R{r}_j=S_3\setminus 23,{\eta }_R{r}_k=1,12,123,{\eta }_R{r}_t={\eta }_R{r}_k\cup 23.\end{array}$$

${\eta }_R$ is a soft 1-absorbing prime ideal, as following conditions are satisfied:

• ${\eta }_R{r}_k-r_t={\eta }_R{r}_k+r_j={\eta }_R{r}_t\supseteq {\eta }_R{r}_k\cap {\eta }_R{r}_t$

• ${\eta }_R{r}_t-r_k={\eta }_R{r}_t+r_k={\eta }_R{r}_j\supseteq {\eta }_R{r}_t\cap {\eta }_R{r}_k$

• ${\eta }_R{r}_t-r_j={\eta }_R{r}_t+r_t={\eta }_R{r}_k\supseteq {\eta }_R{r}_t\cap {\eta }_R{r}_j$

• ${\eta }_R{r}_j-r_t={\eta }_R{r}_j+r_j={\eta }_R{r}_k\supseteq {\eta }_R{r}_j\cap {\eta }_R{r}_t$

• ${\eta }_R{r}_k-r_j={\eta }_R{r}_k+r_t={\eta }_R{r}_j\supseteq {\eta }_R{r}_k\cap {\eta }_R{r}_j$

• ${\eta }_R{r}_j-r_k={\eta }_R{r}_j+r_k={\eta }_R{r}_t\supseteq {\eta }_R{r}_j\cap {\eta }_R{r}_k$

• ${\eta }_R{r}_k{r}_t={\eta }_R{r}_t\supseteq {\eta }_R{r}_k\cup {\eta }_R{r}_t$

• ${\eta }_R{r}_k{r}_t^2={\eta }_R{r}_k{r}_t\cup {\eta }_R{r}_t$

• ${\eta }_R{r}_k^2{r}_t={\eta }_R{r}_k{r}_t\cup {\eta }_R{r}_k$

Since ${\eta }_R{r}_j{r}_k={\eta }_R{r}_i=S_3\ne S_3\setminus 23={\eta }_R{r}_j\cup {\eta }_R{r}_k$, ${\eta }_R$ is not a soft prime ideal.

Proposition 3.

Let $R$ be a nonunitary and commutative ring, and let $R$ have at least one nonzero nilpotent element. $r_j^2,{\eta }_R{r}_j^2\tilde{\in }\text{ZD}{\eta }_R$, if $r_j^2\ne 0$ for all $r_j\in R\setminus 0$, and ${\eta }_R$ is a one-to-one soft $1$-absorbing prime ideal.

Proof 17.

Let $0\ne r_i\in$ Nil $R$. Then, $r_i^n=0$ for some $n\ge 3\in \mathbb{N}$. We see that $r_i^n{r}_j^n=0$ for all $r_j\in R\setminus 0$. Since ${\eta }_R$ is a one to one soft 1-absorbing prime ideal, ${\eta }_{R}0\ne {\eta }_R{r}_i^2$ and ${\eta }_{R}0\ne {\eta }_R{r}_j^2$. Hence,$$\begin{array}{}\text{(36)}& {\eta }_{R}0={\eta }_R{r}_i^n{r}_j^n={\eta }_R{r}_i^2{r}_j^2={\eta }_R{r}_i{r}_i{r}_j^2={\eta }_R{r}_i^2\cup {\eta }_R{r}_j^2⟹r_j^2,{\eta }_R{r}_j^2\tilde{\in }ZD{\eta }_R.\end{array}$$

Proposition 4.

Let $R$ be a nonunitary and commutative ring, and ${\eta }_R$ be a one-to-one soft 1-absorbing prime ideal that is not a soft prime ideal. $r_i,{\eta }_R{r}_i\tilde{\in }ZD{\eta }_R$, if $r_i,{\eta }_R{r}_i\tilde{\in }\text{Nil}{\eta }_R$ and $r_i{r}_j\ne 0$, for all $r_j\in R\setminus 0$.

Proof 18.

Since $r_i,{\eta }_R{r}_i\tilde{\in }\text{Nil}{\eta }_R$, and ${\eta }_R$ is not a soft prime ideal, we obtain that $\sqrt{{\eta }_R}{r}_i={\eta }_{R}0\ne {\eta }_R{r}_i$. For all $r_j\in R\setminus 0$,$$\begin{array}{c}\text{(37)}& {\eta }_R{r}_i^2={\eta }_{R}0,{\eta }_R{r}_i^2\cup {\eta }_R{r}_j={\eta }_{R}0,\\ {\eta }_R{r}_i^2{r}_j={\eta }_{R}0,\\ {\eta }_R{r}_i\cup {\eta }_R{r}_i{r}_j={\eta }_{R}0,\\ r_i,{\eta }_R{r}_i\tilde{\in }ZD{\eta }_R.\end{array}$$

Proposition 5.

Let $R$ be a nonunitary and commutative ring, $n\ge 2\in \mathbb{N}$ and $\mathrm{\Xi }=r_i\in R:{\eta }_R{r}_i{r}_k^n={\eta }_{R}0\text{\hspace{0.17em}for\hspace{0.17em}some\hspace{0.17em}}{r}_k\in R$. $\mathrm{\Xi }$ is an ideal of $R$ if ${\eta }_R$ is a soft 1-absorbing prime ideal.

Proof 19.

It is obvious that $0\in \mathrm{\Xi }$ and $\mathrm{\Xi }\subseteq R$. Let $r_i,r_j\in \mathrm{\Xi }$. Then, ${\eta }_R{r}_i{r}_k^n={\eta }_R{r}_j{r}_l^n={\eta }_{R}0$ for some $r_k,r_l\in R$. It is observed that$$\begin{array}{c}\text{(38)}& {\eta }_R{r}_i-r_j{r}_k^n{r}_l^n={\eta }_R{r}_i{r}_k^n{r}_l^n-r_j{r}_k^n{r}_l^n\supseteq {\eta }_R{r}_i{r}_k^n{r}_l^n\cap {\eta }_R{r}_j{r}_k^n{r}_l^n\\ ={\eta }_R{r}_i{r}_k^n\cup {\eta }_R{r}_l^n\cap {\eta }_R{r}_j{r}_l^n\cup {\eta }_R{r}_k^n\\ ={\eta }_{R}0.\end{array}$$

Thus,$$\begin{array}{c}\text{(39)}& {\eta }_R{r}_i-r_j{r}_k^n{r}_l^n={\eta }_{R}0,{\eta }_R{r}_i-r_j{r_k{r}_l}^n={\eta }_{R}0,\\ r_i-r_j\in \mathrm{\Xi }.\end{array}$$

For all $r\in R$,$$\begin{array}{}\text{(40)}& {\eta }_R{r}_{i}r{r}_k^n={\eta }_R{r}_i{r}_k^{n}r={\eta }_R{r}_i{r}_k^n\cup {\eta }_{R}r={\eta }_{R}0⟹r_{i}r\in \mathrm{\Xi }.\end{array}$$

Since $R$ is a commutative ring, we obtain $r{r}_i=r_{i}r\in \mathrm{\Xi }$. As a result, $\mathrm{\Xi }$ is an ideal of $R$.

Proposition 6.

Let $R$ be a commutative ring, $r_i{r}_j=0$ for all $r_i\ne r_j\in R$, and$$\begin{array}{}\text{(41)}& \daleth =r_i\in R:\sqrt{{\eta }_R}{r}_i={\eta }_{R}0.\end{array}$$

If ${\eta }_R$ is a soft 1-absorbing prime ideal, then $\daleth$ is an ideal of $R$.

Proof 20.

Since ${\eta }_R$ is a soft 1-absorbing prime ideal, $\sqrt{{\eta }_R}{r}_i={\eta }_R{r}_i^2={\eta }_{R}0$ for all $r_i\in R$. It is obvious that $\daleth \ne \varnothing$ and $\daleth \subseteq R$. Let $r_i,r_j\in \daleth$. Then, ${\eta }_R{r}_i^2={\eta }_R{r}_j^2={\eta }_{R}0$. Moreover,$$\begin{array}{c}\text{(42)}& {\eta }_R{r_i-r_j}^2={\eta }_R{r}_i^2-r_i{r}_j-r_i{r}_j-r_j^2\supseteq {\eta }_R{r}_i^2-r_i{r}_j\cap {\eta }_R{r}_i{r}_j-r_j^2\\ \supseteq {\eta }_R{r}_i^2\cap {\eta }_R{r}_i{r}_j\cap {\eta }_R{r}_i{r}_j\cap {\eta }_R{r}_j^2\\ ={\eta }_{R}0,\\ {\eta }_R{r_i-r_j}^2={\eta }_{R}0\\ r_i-r_j\in \daleth .\end{array}$$

Again, since ${\eta }_R$ is a soft 1-absorbing prime ideal, we obtain$$\begin{array}{}\text{(43)}& {\eta }_R{r_{i}r}^2={\eta }_R{r}_i^2{r}^2={\eta }_R{r}_i^2\cup {\eta }_R{r}^2={\eta }_{R}0⟹r_{i}r\in \daleth ,\end{array}$$for all $r\in R$. Since $R$ is a commutative ring, $r_{i}r=r{r}_i\in \daleth$. As a result, $\daleth$ is an ideal of $R$.

Proposition 7.

Let $R$ be a nonunitary and commutative ring, $I$ be an ideal of $R$, and ${\eta }_R$ be a soft 1-absorbing prime ideal. If ${\eta }_R{r}_i=\mathrm{\Gamma }$ for all $r_i\in I\setminus 0$ and ${\eta }_R{r}_j={\mathrm{\Gamma }}^{\mathrm{\ast }}$ for all $r_j\in R\setminus I$, then ${\mathrm{\Gamma }}^{\mathrm{\ast }}\subseteq \mathrm{\Gamma }$.

Proof 21.

Since $I$ is an ideal of $R$, $r_i\in I⟹r_i^2\in I⟹r_i^2{r}_j\in I$ for all $r_j\in R\setminus I$. Hence, ${\eta }_R{r}_i^2{r}_j=\mathrm{\Gamma }$. Since ${\eta }_R$ a soft 1-absorbing prime ideal,$$\begin{array}{}\text{(44)}& {\eta }_R{r}_i^2{r}_J=\mathrm{\Gamma }⟹{\eta }_R{r}_i^2\cup {\eta }_R{r}_j=\mathrm{\Gamma }⟹\mathrm{\Gamma }\cup {\mathrm{\Gamma }}^{\mathrm{\ast }}=\mathrm{\Gamma }⟹{\mathrm{\Gamma }}^{\mathrm{\ast }}\subseteq \mathrm{\Gamma }.\end{array}$$

Proposition 8.

Let $R$ be a nonunitary ring and ${\eta }_R$ be a soft 1-absorbing prime ideal. Let $I$ be an ideal of $R$ such that either $r_i{r}_j\in I$ or $r_k\in I$ when $r_i{r}_j{r}_k\in I$ for $r_i,r_j,r_k\in R$. If ${\eta }_R{r}_i={\eta }_{R}0$ for all $r_i\in I$, then $r_j,{\eta }_R{r}_j\tilde{\in }\text{Nil}{\eta }_R$ for all $r_j\in \sqrt{I}\setminus I$.

Proof 22.

Let $r_j\in \sqrt{I}\setminus I$. Due to the property of $I$, $r_j^2\in I$. From the definition of ${\eta }_R$, we obtain ${\eta }_R{r}_j^2={\eta }_{R}0$. Since ${\eta }_R$ is a soft 1-absorbing prime ideal, it is observed that $\sqrt{{\eta }_R}{r}_j={\eta }_{R}0$. Finally, $r_j,{\eta }_R{r}_j\tilde{\in }\text{Nil}{\eta }_R$.

Theorem 24.

Let $R$ and $S$ be nonunitary rings, $⊺:R⟶S$ be an onto ring homomorphism, and ${\eta }_R$ be a soft 1-absorbing prime ideal. The soft set $⊺{\eta }_R$ is a soft 1-absorbing prime ideal.

Proof 23.

By Theorem 3, we know that $⊺{\eta }_R$ is a soft ideal. It is needed to show the soft 1-absorbing primality of $⊺{\eta }_R$. For all $s_i,s_j,s_k\in S$,$$\begin{array}{c}\text{(45)}& ⊺{\eta }_R{s}_i{s}_j{s}_k=\cup {\eta }_{R}r:r\in R,⊺r=s_i{s}_j{s}_k=\cup {\eta }_R{r}_i{r}_j{r}_k:r_i,r_j,r_k\in R,⊺r_i{r}_j{r}_k=s_i{s}_j{s}_k\\ =\cup {\eta }_R{r}_i{r}_j{r}_k:r_i,r_j,r_k\in R,⊺r_i{r}_j=s_i{s}_j,⊺r_k=s_k\\ =\cup {\eta }_R{r}_i{r}_j\cup {\eta }_R{r}_k:r_i,r_j,r_k\in R,⊺r_i{r}_j=s_i{s}_j,⊺r_k=s_k\\ =\cup {\eta }_R{r}_i{r}_j:r_i,r_j\in R,⊺r_i{r}_j=s_i{s}_j\cup \cup {\eta }_R{r}_k:r_k\in R,⊺r_k=s_k\\ =⊺{\eta }_R{s}_i{s}_j\cup ⊺{\eta }_R{s}_k.\end{array}$$

The proof is completed.

Theorem 25.

Let $R$ and $S$ be nonunitary rings, $⊺:R⟶S$ be a ring homomorphism, and ${\eta }_S$ be a soft 1-absorbing prime ideal of $S$. The soft set ${⊺}^{-1}{\eta }_S$ is a soft 1-absorbing prime ideal.

Proof 24.

By Theorem 3, we know that $⊺{{\eta }_R}^{-1}$ is a soft ideal. It is needed to show the soft 1-absorbing primality of ${⊺}^{-1}{\eta }_S$. For all $r_i,r_j,r_k\in R$,$$\begin{array}{c}\text{(46)}& {⊺}^{-1}{\eta }_S{r}_i{r}_j{r}_k={\eta }_S⊺r_i{r}_j{r}_k={\eta }_S⊺r_i{r}_j⊺r_k\\ ={\eta }_S⊺r_i{r}_j\cup {\eta }_S⊺r_k\\ ={⊺}^{-1}{\eta }_S{r}_i{r}_j\cup {⊺}^{-1}{\eta }_S{r}_k.\end{array}$$

The proof is completed.

Theorem 26.

Let $I$ be an ideal of $R$ and ${\eta }_R$ be a soft 1-absorbing prime ideal. There is a soft ideal of $R/I$, which has the same properties of ${\eta }_R$. The radicals are equal to each other.

Proof 25.

Let ${\chi }_{R/I}$ be a soft ideal of $R/I$ and ${\chi }_{R/I}I+r_i={\eta }_R{r}_i$ for all $r_i\in R$. Since ${\eta }_R$ is a soft 1-absorbing prime ideal, then$$\begin{array}{}\text{(47)}& {\chi }_{R/I}I+r_i-I+r_j={\chi }_{R/I}I+r_i-r_j={\eta }_R{r}_i-r_j\supseteq {\eta }_R{r}_i\cap {\eta }_R{r}_j={\chi }_{R/I}I+r_i\cap {\chi }_{R/I}I+r_j,\end{array}$$and$$\begin{array}{}\text{(48)}& {\chi }_{R/I}I+r_{i}I+r_j={\chi }_{R/I}I+r_i{r}_j={\eta }_R{r}_i{r}_j\supseteq {\eta }_R{r}_i\cup {\eta }_R{r}_j={\chi }_{R/I}I+r_i\cup {\chi }_{R/I}I+r_j,\end{array}$$for all $I+r_i,I+r_j\in R/I$. Here, ${\chi }_{R/I}$ is a soft 1-absorbing prime ideal. We prove the 1-absorbing primality of ${\chi }_{R/I}$,$$\begin{array}{c}\text{(49)}& {\chi }_{R/I}I+r_{i}I+r_{j}I+r_k={\chi }_{R/I}I+r_i{r}_j{r}_k={\eta }_R{r}_i{r}_j{r}_k\\ ={\eta }_R{r}_i{r}_j\cup {\eta }_R{r}_k\\ ={\chi }_{R/I}I+r_{i}I+r_j\cup {\chi }_{R/I}I+r_k,\end{array}$$for all $I+r_i,I+r_j,I+r_k\in R/I$. It is gained that if ${\eta }_R$ is a soft prime or soft near-prime ideal, then ${\chi }_{R/I}$ is a soft prime ideal or soft near-prime ideal. Lastly, we prove that $\sqrt{{\chi }_{R/I}}=\sqrt{{\eta }_R}$. Thus,$$\begin{array}{}\text{(50)}& \sqrt{{\chi }_{R/I}}I+r_i=\underset{l=1}{\overset{n}{\cup }}{\chi }_{R/I}{I+r_i}^l=\underset{l=1}{\overset{n}{\cup }}{\chi }_{R/I}I+r_i^l=\underset{l=1}{\overset{n}{\cup }}{\eta }_R{r}_i^l=\sqrt{{\eta }_R}{r}_i,\end{array}$$for all $r_i\in R$.

6. Aim and Contribution

The main contributions of this article are as follows:

• The concepts of soft nilpotents, soft idempotents, and soft zero divisors are defined, and their behaviors on soft prime int-ideals are investigated

• Two different generalizations of soft prime int-ideals are obtained using a unitary ring $R$

• Through the soft algebraic concepts defined in this study, it is emphasized that the relationship between soft algebraic structures and classical algebraic structures is very strong

7. Conclusions

This paper presents the properties of soft prime int-ideal, soft near-prime int-ideal, and soft 1-absorbing prime int-ideal. First, we demonstrated that soft prime int-ideals and prime ideals can be related. Then, we examined soft near-prime int-ideal and soft 1-absorbing prime int-ideals as generalizations of soft prime int-ideals. Furthermore, we showed that if $R$ is a commutative ring and ${\eta }_R$ is a soft near-prime int-ideal, then $\sqrt{{\eta }_R}$ is a soft prime int-ideal. Finally, we verified that the properties of soft prime int-ideals and soft near-prime int-ideals are preserved under ring homomorphisms. In the future, new types of soft int-ideals, such as soft primary int-ideals, soft 1-absorbing primary int-ideals, soft 2-absorbing int-ideals, and soft 2-absorbing primary int-ideals, could be defined. The relationships between these new ideals can be explored, and their algebraic features can be studied. In addition, a weakly soft 1-absorbing prime int-ideal can be defined by modifying the definition of soft 1-absorbing prime int-ideals to require $r_i{r}_j{r}_k\ne 0$, and its algebraic properties can be investigated.

Funding

This work received no external funding.

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