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Notes on annihilator conditions in modules over commutative rings. (English) Zbl 1224.13002

Summary: Let \(M\) be a module over the commutative ring \(R\). In this paper we introduce two new notions, namely strongly coprimal and super coprimal modules. Denote by \(Z_R(M)\) the set of all zero-divisors of \(R\) on \(M\). \(M\) is said to be strongly coprimal (resp. super coprimal) if for arbitrary \(a,b\in Z_R(M)\) (resp. every finite subset \(F\) of \(Z_R(M)\)) the annihilator of \(\{a,b\}\) (resp. \(F\)) in \(M\) is non-zero. In this paper we give some results on these classes of modules. We also provide a relationship between the families of coprimal, strongly coprimal and super coprimal modules. We prove that if \(M\) is a coprimal module of finite Goldie dimension over a commutative ring, then \(M\) is super coprimal. Finally we show that every proper submodule of a module over a Prüfer domain of finite character can be expressed as a finite intersection of strongly primal submodules.

MSC:

13A05 Divisibility and factorizations in commutative rings
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
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