Primal, completely irreducible, and primary meet decompositions in modules
Inspired by work of \textit{L. Fuchs}, \textit{W. Heinzer} and \textit{B. Olberding} [Trans. Am. Math. Soc. 357, No. 7, 2771--2798 (2005; Zbl 1066.13003); ibid. 358, No. 7, 3113--3131 (2006; Zbl 1091.13011)] on primal and completely irreducible ideals , this paper studies similar and related concepts for modules. A proper submodule \(N\) of \(M\) is called (meet) irreducible if \(N=K\cap L\) for submodules \(K\) and \(L\) of \(M\), implies that \(N=K\) or \(N=L\). A proper submodule \(N\) of \(M\) is said to be completely irreducible (CI) if \(N\) is not the intersection of any collection of submodules of \(M\) each properly containing \(N\). A module \(M\) is called coirreducible (resp. completely coirreducible) if 0 is an irreducible (CI) submodule of \(M\). A proper submodule \(N\) of an \(R\)-module \(M\) over a commutative ring is said to be a primary submodule of \(M\) if \(rm\in N\) for \(r\in\), \(m\in M\), implies that \(m\in N\) or \(r\in R_m (N)\) (\(=\{r\in R|\exists k\in\mathbb{R}{\text{ such that }}r^k M\subseteq N\}\)). For a few instances, such as in the case where \(M\) is a module with primary decomposition, if \(M\) is a Laskerian module or if \(R\) has classical Krull dimension 0, it is shown that every CI submodule of \(M\) is \(\mathbf m\)-primary for some \({\mathbf m}\in \mathrm{Max}(R)\). In the case of a Noetherian ring, a submodule of \(M\) is CI if and only if it is an irreducible \(\mathbf p\)-primary submodule for some \({\mathbf p}\in \mathrm{Max}(R)\).NEWLINENEWLINE A submodule \(N\) of \(M\) over a commutative ring \(R\) is said to be primal if \(N\neq M\) and the set \(Z(M/N)\) of all zero divisors of \(M/N\) is an ideal of \(R\); \(M\) is called coprimal if \(0\) is a primal submodule of \(M\). For a Noetherian ring \(R\), several equivalences are found for every primal submodule (resp. ideal) of \(R\) to be primary.NEWLINENEWLINE Let \(A\leq C\) be submodules of \(M_R\). Then \(A\) is a relevant completely irreducible divisor (RCID) of \(A\) if \(A\) has a decomposition as an intersection of completely irreducible submodules of \(M\) in which \(C\) appears and is relevant. Following \textit{J. Fort} [Math. Z. 103, 363--388 (1968; Zbl 0155.07501)], \(M\) is said to be rich in coirreducibles (RC) if \(M \neq 0\) and each of its non-zero submodules contains a coirreducible submodule. An RC module is then characterized in terms of essential extensions and its injective hull. \textit{T. Albu} and \textit{P. F. Smith} [Rev. Roum. Math. Pures Appl. 54, No. 4, 275--286 (2009; Zbl 1199.13011)] defined a nonzero module \(M\) to be RCC if every nonzero submodule of \(M\) contains a simple submodule. A module \(M\) is shown to be RCC iff the socle of \(N\) is essential in \(M\) iff M is an essential extension of a direct sum of simple submodules of \(M\), to mention but a few of the equivalent conditions that are found. The set of zero divisors of \(M/N\) is described for the case where \(N\) is an irreducible intersection of CI submodules. This leads to the result that if \(N=\cap N_i\) is an irredundant intersection of CI submodules of a module \(M\), then \(N\) is \(\mathbf m\)-primal iff each \(N_i\) is \(\mathbf m\)-primal, \({\mathbf m}\in \mathrm{Max}(R)\). The paper concludes with a list and short discussion of seven open problems.
scientific article; zbMATH DE number 6090192