Residually small commutative rings

Abstract

Let $R$ be a ring. Following the literature, $R$ is called \textit {residually finite} if, for every $r\in R\backslash \{0\}$, there exists an ideal $I_r$ of $R$ such that $r\notin I_r$ and $R/I_r$ is finite. In this note, we define a \textit {strictly} infinite commutative ring $R$ with identity to be \textit {residually small} if, for every $r\in R\backslash \{0\}$, there exists an ideal $I_r$ of $R$ such that $r\notin I_r$ and $|R/I_r|\lt |R|$. The purpose of this article is to study such rings, extending results on (infinite) residually finite rings.