Residually small commutative rings
Well established in the literature is a property called residually finite which originated with groups, and since has been studied in modules and rings. A commutative ring \(R\) is called residually finite if for every \(r \in R\setminus \{0\}=R^*\), there exists an ideal \(I_r\) such that \(r\notin I_r\) and \(R/I_r\) is finite. This paper studies a related property of commutative rings called residually small. An infinite commutative ring \(R\) is said to be residually small if for every non-zero \(r \in R\), there exists and ideal \(I_r\) such that \(r\notin I_r\) and \(|R/I_r|< |R|\). It is worth noting that all infinite residually finite rings are therefore also residually small with this definition. This property is related to a property for modules which is called homomorphically smaller (HS) which means for an infinite module \(M\), \(|M/N|<|M|\) for every non-zero submodule \(N\) of \(M\). Then a ring \(R\) is HS if it is an HS module over itself. The paper begins with an extensive preliminary section which explores many of the properties of residually small rings and especially compares these properties with other known properties in the literature. Then significant attention is turned towards studying how the residually small property behaves with respect to polynomial ring extensions, direct products, quotient rings, factor rings and integral extensions. Finally, the authors conclude by posing several interesting open questions of which we include a few. (1) If \(R\) is a Noetherian local ring and \(I\) is an ideal of \(R\) with \(|R/I|=|R|\), then \(R\) is residually small if and only if \(R/I\) is residually small. Does this still hold if \(R\) is not Noetherian? (2) Suppose that \(D_1 \subseteq D_2\) is an integral extension of domains and that \(D_1\) is residually small and uncountable. Must \(D_2\) be residually small? Is this question even decidable in ZFC? What if the extension were finite?
scientific article; zbMATH DE number 6917493
scientific article