On commutative Noetherian rings which have the s. p. a. r. property (Q1383561)

Let \(R\) be a commutative Noetherian ring and \(M\) be an \(R\)-module. A proper submodule \(P\) of \(M\) is prime if \(rm\in P\Rightarrow m\in P\) or \(rM\subseteq P\) and a proper submodule \(N\) of \(M\) is semiprime if, for all positive integers \(k\), \(r^km\in N\Rightarrow rm\in N\). The s.p.a.r. property for \(R\) is that, for every \(R\)-module \(M\), every semiprime submodule of \(M\) is an intersection of prime submodules. Let \(P_1,P_2,\ldots,P_n\) be the minimal prime ideals of \(R\). The main result of this paper is that \(R\) has the s.p.a.r. property if and only if the following conditions hold: (i) \(R\) has Krull dimension \(1\) and, for \(1\leq k\leq n\), \(R/P_k\) is a Dedekind domain; (ii) if \(n\geq 2\), for \(1\leq k\leq n-1\), \((\bigcap_{i=1}^{k} P_i)+P_{k+1}= \bigcap_{i=1}^{k}(P_i+P_{k+1})\); (iii) if \(n\geq 2\), for \(1\leq i<j\leq n\), \(R/(P_i+P_j)\) is semisimple Artinian or zero. This generalises a result of \textit{J. Jenkins} and \textit{P. F. Smith} [Commun. Algebra 20, No. 12, 3593-3602 (1992; Zbl 0776.13003)], for the case where \(R\) is a domain of finite global dimension.