Nil-prime ideals of a commutative ring
Faranak Farshadifar
TL;DR
This work proposes nil-prime ideals as a nil-version generalization of prime ideals in commutative rings with identity, defined via the existence of a witness $x \in Nil(R)$ such that $ab \in P$ forces $a \in P$, $b \in P$, $a+x \in P$, or $b+x \in P$. It clarifies the relationship of nil-prime ideals to prime and $\mathfrak{N}$-prime ideals and develops nil-versions of several ring-theoretic concepts, including nil-maximal, nil-minimal, and nil-principal ideals, as well as $\mathfrak{N}(R)$-integral domains and $\mathfrak{N}(R)$-PIDs. Key results show that $Nil(R) \subseteq P$ implies $P$ is prime, and $Nil(R)=0$ yields equality between nil-prime and prime ideals; the paper also analyzes behavior under quotients, polynomial extensions, and products, and establishes structural results in Artinian rings (e.g., $\mathfrak{N}$-prime ideals are $\mathfrak{N}$-maximal with finitely many nil-distinct nil-maximal ideals). Overall, the findings provide a coherent framework for nil-extensions in ring theory with potential geometric implications.
Abstract
Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P of R is a nil-prime ideal if there exists x \in N(R) and whenever ab \in P, then a \in P or b \in P or a+x \in P or b+x \in P for each a,b \in R. Also, we introduce nil versions of some algebraic concepts in ring theory such as nil-maximal ideal, nil-minimal ideal, nil-principal ideal and investigate some nil-version of a well-known results about them.