An equivalence of categories

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2004-01-01
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Crabtree, David
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Mathematics
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In this paper, we first explore an interesting construction called the split extension. The split extension is a way of constructing another ring from an arbitrary ring R and an R-module M. This split extension gives us a nice way of obtaining a new ring from an R-module. It is this correspondence that will be exploited in the paper. One of the other primary concepts appearing in this paper is the notion of an abelian group in a category, or more generally, a group in a category. The diagram version of the definition of a group in a category is merely a specific instance of a more general concept; namely, the concept of an "interpretation of an equational algebraic theory" in a category. In particular, the notion of a group in a category arises via an interpretation of the equational theory of groups. The paper also contains an appendix which explores the equivalence of two different definitions of an abelian group in a category (the "diagram" definition, and "functor" definition).

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Thu Jan 01 00:00:00 UTC 2004