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Principally dual stable modules

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 اسعد محمد علي حسين الحسيني
01/04/2013 17:36:46
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In two previous papers ([2] and [3]), we introduced the concept of fully d-stable modules and studied some generalizations of it. A submodule of an -module is said to be d-stable if for every homomorphism , the module is said to be fully d-stable, if each of its submodules is d-stable [2]. Full d-stability is dual to the concept of full stability introduced by ABBS in [1], and both of these concepts are stronger than duo property of modules. A submodule of an -module is said to be stable if , for any homomorphism ,
a module is fully stable of all of its submodules are stable [1]. In [1], it was proved that a module is fully stable if and only if each cyclic submodule is stable. Unfortunately it is not the case in full d-stability. This motivates introducing the concept of principally d-stable module which is a generalization of full d-stability. A module will be called principally d-stable if every cyclic submodule of it is d-stable. In this paper we studied this new concept and the conditions that make a principally d-stable module into a fully d-stable. In section 2 main properties of principal d-stable were investigated in addition, we see that quasi-projectivity is a sufficient condition for a principal d-stable module to be fully d-stable. Also we show that over Dedekind domain and integral domain with certain conditions, the two concepts, full (and principal) d-stability coincide. Links between the two dual concepts full stability and full d-stability, in certain conditions, also, was found . In section 3, under regular modules (in some sense), many characterizations to principally d-stable module, via endomorphism rings, were investigated.


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  • fully(principally) d-stable module;quasi-projective, duo, regular module