## 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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