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On the prime submodules of multiplication modules. (English) Zbl 1042.16001

Throughout the paper \(R\) denotes a commutative ring with identity and all modules are unitary \(R\)-modules. Let \(M\) be a module. A proper submodule \(K\) of \(M\) is called prime if \(rm\in K\), for \(r\in R\) and \(m\in M\), then \(r\in(K:M)=\{r\in R\mid rM\subseteq K\}\). \(M\) is called a multiplication module if for every submodule \(N\) of \(M\), there is an ideal \(I\) of \(R\) such that \(N=IM\). For a proper submodule \(N\) of \(M\), the radical of \(N\), denoted \(M\text{-rad}(N)\), is the intersection of all prime submodules of \(M\) containing \(N\). For a submodule \(N\) of \(M\) such that \(N=IM\) for some ideal \(I\) of \(R\), \(I\) is said to be a presentation (ideal) of \(N\). An element \(u\in M\) is said to be a unit provided \(u\) is not contained in any maximal submodule of \(M\). The author gives the following definition: if \(N=IM\) and \(K=JM\) for some ideals \(I\) and \(J\) of \(R\), the product of \(N\) and \(K\) is defined by \(IJM\) and is denoted by \(NK\). For a multiplication module \(M\) and \(m,m'\in M\), the product of \(Rm\) and \(Rm'\) is denoted by \(mm'\).
The author establishes the following main results: (1) If \(N=IM\) and \(K=JM\) are submodules of a multiplication module \(M\), then \(NK\) is independent of presentations of \(N\) and \(K\); (2) If \(N\) is a submodule of a multiplication module \(M\), then \(M\text{-rad}(N)=\{m\in M\mid m^k\subseteq N\) for some \(k\geq 0\}\); (3) Let \(M\) be a faithful multiplication module such that \(M\) has a unit \(u\). Then for every submodule \(N\) of \(M\), the following conditions are equivalent: (i) \(N\) is contained in every maximal submodule of \(M\); (ii) \(u-rx\) is a unit for all \(r\in R\) and for all \(x\in N\); (iii) if \(M\) is finitely generated and \(NM=M\), then \(M=0\); (iv) if \(M\) is finitely generated and \(K\) is a submodule of \(M\) such that \(M=NM+K\), then \(M=K\).

MSC:

16D80 Other classes of modules and ideals in associative algebras
16D10 General module theory in associative algebras
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