On commutative rings without prime ideals

Abstract

In commutative ring theory, the prevalence of prime ideals in a ring often reveals underlying algebraic structure in the ring. This thesis is dedicated to studying rings containing no prime ideals, which the author terms null spec rings. This thesis gives examples of null spec rings and proves that having no prime ideals is equivalent to being a nil ring. The author also proves that a ring is null spec if and only if any integral extension of the ring is null spec if and only if the polynomial ring with coefficients in the null spec ring is null spec. Well-known generalizations of "prime ideal" are considered in the context of null spec rings.