On regular rings whose cyclic faithful modules contain generators (Q1371121)

For a ring \(R\), \(Q(R)\) denotes the right maximal ring of quotients of \(R\). A ring \(R\) is said to satisfy the condition (C) if every cyclic faithful right module contains a submodule which is a generator for Mod-\(R\). In this paper, von Neumann regular rings with the condition (C) are investigated. For a von Neumann regular ring \(R\), it is shown that \(R\) satisfies (C) if and only if \(R\cong\prod^k_{i=1}\text{Mat}_{n(i)}(S_i)\), where \(n(1)=1\), and \(n(i)\geq 2\) for \(i=2,3,\dots,k\), and where each \(S_i\) is an abelian regular ring such that for \(i=2,3,\dots,k\), every finitely generated faithful right \(S_i\)-submodule of \(Q(S_i)\) contains a unit in \(Q(S_i)\). Moreover, with replacing the condition ``cyclic'' by the condition ``finitely generated'', it is shown that, for a von Neumann regular ring \(R\), every finitely generated faithful right \(R\)-module contains a submodule which is a generator for Mod-\(R\) if and only if \(R\cong\prod^k_{i=1}\text{Mat}_{n(i)}(S_i)\), where each \(S_i\) is an abelian regular ring such that every finitely generated faithful right \(S_i\)-submodule of \(Q(S_i)\) contains a unit in \(Q(S_i)\). Several interesting examples are also provided.