Centrally Essential Rings and Semirings

Abstract

This work is a review of results about centrally essential rings and semirings. A ring (resp., semiring) is said to be centrally essential if it is either commutative or satisfy the property that for any non-central element a, there exist non-zero central elements x and y with ax = y. The class of centrally essential rings is very large; many corresponding examples are given in the work.non-central element a ∈ A, there are non-zero central elements x and y with ax = y.It is clear that any commutative ring is centrally essential. An unital ring A with center C = Z(A) is centrally essential if and only if the C-module A is an essential extension of C C .From the definition of a centrally essential ring A, it might seem that such a ring is possibly commutative. Indeed, A satisfies many propeties of commutative rings. For example,• all idempotents of A are central, see 1.1.4 below;• If A is a centrally essential local ring, then the ring A/J(A) is a field and, in particular, is commutative; see 1.3.2.• If A is a right or left semi-Artinian centrally essential ring, then the factor ring A/J(A) is commutative; see 1.4.5.However, a centrally essential ring A may be very far from a commutative ring. For example,