Finitely-generated modules over a principal ideal domain
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چکیده
Let R be a commutative ring throughout. Usually R will be an integral domain and even a principal ideal domain, but these assumptions will be made explicitly. Since R is commutative, there is no distinction between left, right and 2-sided ideals. In particular, for every ideal I we have a quotient ring R/I. F always denotes a field. Our goal is to prove the classification theorem for finitely-generated modules over a principal ideal domain, which comes in two versions: elementary divisors and invariant factors. For R = Z this gives a classification of finitely-generated abelian groups, while for R = F [x] it can be use to classify n× n-matrices up to similarity.
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