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Residual smallness in commutative algebra

ABSTRACT In Oman and Salminen [19], the authors introduce and study residually small rings, defined as follows: an infinite commutative ring R with identity is residually smallif for every r∈R∖{0}, there exists an ideal Ir of R such that r∉Ir and |R∕Ir|<|R|. The purpose of this note is to extend our study. In particular, we continue our investigation of residually small rings and then generalize this notion to modules.