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.