On modules \(M\) for which \(N\cong M\) for every submodule \(N\) of size \(|M|\) (Q847989)

Let \(R\) be a commutative ring with identity. A well-known class of \(R\)-modules is constituted by the so-called \textit{Jonsson modules}, that is the infinite \(R\)-modules \(M\) such that, for every submodule \(N\) of \(M\), \(\mid N\mid=\mid M\mid\) implies that \(N=M\). This paper deals with a natural generalization of this notion, namely with the infinite \(R\)-modules \(M\) such that, for every submodule \(N\) of \(M\), \(\mid N\mid=\mid M\mid\) implies that \(N\cong M\). Such modules are called congruent modules. Their structure is known if \(R\) is the ring of integers or a Dedekind ring. In the current paper congruent modules over an arbitrary commutative ring are studied.