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Derivation modules of free joins and $ m$-adic completions of algebras


Author: I. Y. Chung
Journal: Proc. Amer. Math. Soc. 34 (1972), 49-56
MSC: Primary 13B10
DOI: https://doi.org/10.1090/S0002-9939-1972-0296061-6
MathSciNet review: 0296061
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Abstract: A free commutative join of subalgebras corresponds to a direct sum of submodules in a universal derivation module. In particular, indeterminates of a polynomial ring correspond to elements of a linearly independent set in a universal derivation module. As an application, a simple proof of the uniqueness of cardinalities of indeterminates of a polynomial ring can be obtained by using that of linear bases of a free module over a commutative ring. Similar observations are made for $ \mathfrak{m}$-adic completions of algebras and their derivation modules. Also, the module of linear differential forms of an $ \mathfrak{m}$-adic completion of an algebra is studied.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0296061-6
Keywords: Free commutative join of algebras, $ \mathfrak{m}$-adic completion, polynomial ring, ring of formal power series, tensor product of algebras, universal derivation module of an algebra
Article copyright: © Copyright 1972 American Mathematical Society