Properties of domainlike rings

Abstract

In this paper we will examine properties of and relationships between rings that share some properties with integral domains, but whose definitions are less restrictive. If $R$ is a commutative ring with identity, we call $R$ a \textit{domainlike} ring if all zero-divisors of $R$ are nilpotent, which is equivalent to $(0)$ being primary. We exhibit properties of domainlike rings, and we compare them to presimplifiable rings and (hereditarily) strongly associate rings. Further, we consider idealizations, localizations, zero-divisor graphs, and ultraproducts of domainlike rings.

Associate, Hyperdomainlike, and Presimplifiable Hyperrings

“…In this section, we will provide some definitions that will be used in the sequel. For full details about associate, domainlike, and presimplifiable rings, the reader should see [1,[4][5][6]12]. Also, for details about hyperstructures and hyperrings, the reader should see [12].…”

“…(3) is called a presimplifiable ring if, for any two elements , ∈ with = , we have = 0 or ∈ ( ). Example 3 (see [6]). It is easy to check that Z is presimplifiable if and only if = , where is some prime.…”

“…The study of strongly associate rings began with Kaplansky in [1] and was further studied in [2][3][4][5]. Domainlike rings and their properties were presented by Axtell et al in [6]. Presimplifiable rings were introduced by Bouvier in the series of papers [7][8][9][10][11] and were later studied in [2][3][4].…”