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MONOMORPHISMS AND KERNELS IN THE CATEGORY OF FIRM MODULES

Published online by Cambridge University Press:  24 June 2010

JUAN GONZÁLEZ-FÉREZ
Affiliation:
Department of Applied Mathematics, University of Murcia, 30071 Murcia, Spain
LEANDRO MARÍN
Affiliation:
Department of Applied Mathematics, University of Murcia, 30071 Murcia, Spain
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Abstract

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 : RRMM given by rmrm 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.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

1.Faith, D., Algebra, I. Rings, modules and categories, in Die Grundlehren der mathematischen Wissenschaften, vol. 190. (Springer-Verlag, Berlin, Heidelberg, New York, 1973).Google Scholar
2.García, J. L. and Marín, L., Morita theory for associative rings. Commun. Algebra 29 (12), (2001), 57355856.Google Scholar
3.González-Férez, J. and Marín, L., The category of firm modules need not be abelian. J. Algebra 318 (1) (2007), 377392.CrossRefGoogle Scholar
4.Marín, L., The construction of a generator for R-DMod. in Lecture notes in pure and applied mathematics vol. 210 (New York, Marcel-Dekker, 2000), 287296.Google Scholar
5.Quillen, D. G., Module theory over nonunital rings. 1997 unpublished notes.Google Scholar

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