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Proceedings of the American Mathematical Society

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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
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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Keywords: Free commutative join of algebras, <!– MATH $\mathfrak {m}$ –> <IMG WIDTH="21" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\mathfrak {m}$">-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