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We refer to those injective modules that determine every left exact preradical and that we called main injective modules in a preceding article, and we consider left main injective rings, which as left modules are main injective modules. We prove some properties of these rings, and we characterize QF-rings as those rings which are left and right main injective. 相似文献
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Radoslav M. Dimitrić 《Acta Appl Math》2008,100(2):105-112
This paper generalizes a number of results obtained by Dimitrić in (Glas. Mat. 21(41):327–329, 1986; Proceedings of Hobart Conference on Rings, Modules and Radicals 1987, 204:41–50, Gordon and Breach, 1989) and Dimitrić and Goldsmith in (Glas. Mat. 23(43):241–246, 1988). The original papers were restricted to the category of Abelian groups and orthogonality was to the group of integers ℤ.
Here, we are in a general Abelian category with products and coproducts, with applications to module categories and further
to modules over PID’s. Another generalization is in replacing ℤ by an entire class of subobjects
of the underlying category. We examine properties of the torsion class
, Hom(T,C)=0} in relation to purity, direct summands and indecomposability as well as commutation with direct products, for example.
Of special interest are members of this class when
is a class of slender objects in the ground category; in this case, members of
are called ortho-slender objects. In a sense, ortho-slenderness represents complementary, if not dual, notion to slenderness.
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