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In the case of finitely generated modules over a Dedekind domain, we show that we can even characterize when a surjective or splitting capacity is equal to a fixed nonnegative integer. We generalize Serre, De Stefani-Polstra-Yao, and Bass in a second way by studying the number of summands isomorphic to one module that can appear in a quotient of another module.
We generalize all of these results by replacing the free summand with a direct sum of copies of a finitely presented module.
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This allows us to extend cancellation theorems by Bass and De Stefani-Polstra-Yao, which provide criteria for when two modules with isomorphic direct-sum complements in a larger module are isomorphic.
The complement in each of these theorems is assumed to be finitely generated and projective, although each theorem easily reduces to the case in which the complement is a rank-one free module.
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