Notes on the union of weakly primary submodules (Q410671)

Prime ideals play an important role in commutative ring theory. Let \(R\) be a commutative ring. Of course a proper ideal \(P\) of \(R\) is said to be a prime ideal if \(ab\in P\) implies that \(a\in P\) or \(b\in P\) where \(a, b\in R\). \textit{D. D. Anderson} and \textit{E. Smith} in [Houston J. Math. 29, 831--840 (2003; Zbl 1086.13500)] studied the concept of weakly prime ideals of a commutative ring, where a proper ideal \(P\) of \(R\) is weakly prime if \(a, b\in R\) and \(0\neq ab\in P\) imply that either \(a\in P\) or \(b\in P\). Another generalization of prime ideals is the concept of almost prime ideals. Let \(R\) be an integral domain. As defined in [\textit{S. M. Bhatwadekar} and \textit{P. K. Sharma}, Commun. Algebra 33, 43--49 (2005; Zbl 1072.13003)] a proper ideal \(I\) of \(R\) is said to be almost prime provided that \(a, b\in R\) with \(ab\in I - I^2\) imply that \(a\in I\) or \(b\in I\). This definition can obviously be made for any commutative ring \(R\). A number of generalizations of prime ideals in commutative rings can be found in [\textit{D. D. Anderson} and \textit{M. Bataineh}, Commun. Algebra 36, 686--696 (2008; Zbl 1140.13005)].NEWLINENEWLINEThis was a motivation for several authors to generalize the concept of prime and primary submodules in several ways. Let \(R\) be a commutative ring with identity, and let \(M\) be an \(R\)-module. A proper submodule \(N\) of \(M\) is said to be prime (resp., weakly prime) if \(rm\in N\) (resp., \(0\neq rm\in N\)) implies that either \(m\in N\) or \(rM\subseteq N\) where \(r\in R\) and \(m\in M\). A proper submodule \(N\) of \(M\) is said to be primary (resp., weakly primary) if \(rm\in N\) (resp., \(0\neq rm\in N\)), then either \(m\in N\) or \(r^nM\subseteq N\) for some positive integer \(n\), where \(r\in R\) and \(m\in M\). It is clear that every primary submodule is weakly primary. However, since \(0\) is always weakly primary (by definition), a weakly primary submodule need not be primary.NEWLINENEWLINEIn the paper under review, the authors study weakly primary submodules, and investigate the union of weakly primary submodules of an \(R\)-module \(M\).