Notes on Annihilator Conditions in Modules over Commutative Rings
نویسنده
چکیده
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 ZR(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 ∈ ZR(M) (resp. every finite subset F of ZR(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. Also we 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.
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