## On the spectrum of a module over a commutative ring.(English)Zbl 0876.13002

Let $$R$$ be a commutative ring and $$M$$ an $$R$$-module. A submodule $$N \subset M$$ is prime if $$N\neq M$$ and whenever $$r\in R$$, $$m\in M$$ satisfy $$rm\in N$$ then $$r \in (M:N)$$ or $$m\in N$$. $$M$$ is called a multiplication module if, for each submodule $$P\subset M$$, there exists an ideal $$I\subset R$$ such that $$P=IM$$. If $$M$$ is finitely generated then the spectrum of $$M$$ has a Zariski topology if and only if $$M$$ is a multiplication module.

### MSC:

 13A05 Divisibility and factorizations in commutative rings 13A15 Ideals and multiplicative ideal theory in commutative rings 14A05 Relevant commutative algebra
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### References:

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