Distributive Invariant Centrally Essential Rings

TL;DR

This paper characterizes a broad class of centrally essential rings that are right distributive and invariant, extending known descriptions to include rings with specific radical and factor properties, linking them to well-understood ring types.

Contribution

It generalizes the description of right Noetherian, right distributive centrally essential rings to a larger class, connecting their structure to commutative Pr"ufer domains and Artinian uniserial rings.

Findings

A ring with prime radical that is right distributive and invariant centrally essential decomposes into a product of specific ring types.

Such rings have finitely generated prime radicals and their quotients lack infinite direct sums of non-zero ideals.

The structure theorem applies to rings with particular radical and factor properties, broadening previous classifications.

Abstract

In recent years, centrally essential rings have been intensively studied in ring theory. In particular, they find applications in homological algebra, group rings, and the structural theory of rings. The class of essentially central rings strongly extends the class of commutative rings. For such rings, a number of recent papers contain positive answers to some important questions from ring theory that previously had positive answers for commutative rings and negative answers in the general case. This work is devoted to a similar topic. A familiar description of right Noetherian, right distributive centrally essential rings is generalized on a larger class of rings. Let $A$ be a ring with prime radical $P(A)$. It is proved that $A$ is a right distributive, right invariant centrally essential ring and $P(A)$ is a finitely generated right ideal such that the factor-ring $A/P(A)$ does non contain an infinite direct sum of non-zero ideals if and only if $A=A_1\times\cdots\times A_n$, where every ring $A_k$ is either a commutative Pr\"ufer domain or an Artinian uniserial ring.