Residually Finite Rings

Abstract

Throughout this paper a ring will always be an associative, not necessarily commutative ring with an identity. It is tacitly assumed that the identity of a subring coincides with that of the whole ring. A ring R is said to be residually finite if it satisfies one of the following equivalent conditions:(1) Every non-zero ideal of R is of finite index in R;(2) For each non-zero ideal A of R, the residue class ring R/A is finite;(3) Every proper homomorphic image of R is finite.The class of residually finite rings is large enough to merit our investigation. All finite rings and all simple rings are trivially residually finite. Other residually finite rings are said to be proper.