A characterization of prime submodules
Let \(R\) be a commutative ring (with non-zero identity) and let \(M\) be an \(R\)-module. A submodule \(N\) of \(M\) is called prime if \(N\neq M\) and for each \(a\in R\), \(m\in M\), then \(am\in N\) implies \(m\in N\) or \(aM\subseteq N\). Hence if \(N\) is a prime submodule of \(M\), then the annihilator \(\mathfrak p\) of \(M/N\) is a prime ideal of \(R\) [ cf. \textit{R. L. McCasland} and \textit{M. E. Moore}, Commun. Algebra 20, No. 6, 1803-1817 (1994; Zbl 0776.13007) and \textit{R. L. McCasland} and \textit{P. F. Smith}, Rocky Mt. J. Math. 23, No. 3, 1041-1062 (1993; Zbl 0814.16017)]. The authors obtain interesting results on the relation between prime submodules of \(M\) and prime ideals of \(R\). They prove that to every prime submodule of \(M\) there corresponds a prime ideal of \(R\) and a set of linear equations of a certain type, and conversely.
scientific article; zbMATH DE number 1271831
scientific article