MONOMORPHISMS AND KERNELS IN THE CATEGORY OF FIRM MODULES

In this paper we consider for a non-unital ring R, the category of firm R-modules for a non-unital ring R, i.e. the modules M such that the canonical morphism μM : R ⊗RM → M given by r ⊗ m ↦ rm is an isomorphism. This category is a natural generalization of the usual category of unitary modules for...

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Bibliographic Details
Published in:Glasgow mathematical journal 2010-07, Vol.52 (A), p.83-91
Main Authors: GONZÁLEZ-FÉREZ, JUAN, MARÍN, LEANDRO
Format: Article
Language:English
Subjects:
Categories
Isomorphism
Kernels
Lattice theory
Lattices
Mathematical analysis
Mathematics
Modular
Modules
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Summary:In this paper we consider for a non-unital ring R, the category of firm R-modules for a non-unital ring R, i.e. the modules M such that the canonical morphism μM : R ⊗RM → M given by r ⊗ m ↦ rm is an isomorphism. This category is a natural generalization of the usual category of unitary modules for a ring with identity and shares many properties with it. The only difference is that monomorphisms are not always kernels. It has been proved recently that this category is not Abelian in general by providing an example of a monomorphism that is not a kernel in a particular case. In this paper we study the lattices of monomorphisms and kernels, proving that the lattice of monomorphisms is a modular lattice and that the category of firm modules is Abelian if and only if the composition of two kernels is a kernel.
ISSN:0017-0895
1469-509X
DOI:10.1017/S0017089510000224