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.


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