Tensor products of compact rings
by
Daciana Alina Alb
Department of Mathematics, University of Oradea, Romania
Coauthors: Ursul MIhail (Department of Mathematics, University of Oradea, Romania)

We introduce the notion of a tensor product of compact \Phi-modules over a discrete commutative ring \Phi with identity. Using this notion we prove the existence of tensor products in the category of compact zero-dimensional rings.

Definition. Let \Phi be a commutative discrete ring with identity and A₁, ..., A_n compact \Phi-modules. A pair ( C, \pi) , where C is a compact \Phi-module and \pi is a n-linear mapping of A₁×...×A_n in C is called a tensor product of A₁, ..., A_n provided for every n-linear mapping f:A₁×...×A_n --> D in a compact \Phi-module D there exists a unique continuous \Phi-homomorphism [^f]:C --> D such that \alpha = [^f] o \pi.

Theorem 1. If A₁, ..., A_n are compact unitary \Phi-modules then there exists the tensor product A₁\otimes...\otimesA_n.

Theorem 2. If \Phi is a commutative discrete ring with identity, A, B, C some unitary compact zero-dimensional \Phi-modules, then there exists a unique topological \Phi-isomorphism ( A\otimesB) \otimesC --> A\otimesB\otimesC for which ( a\otimesb) \otimesc --> a\otimesb\otimesc.

Theorem 3. If A and B are two zero-dimensional \Phi-algebras over a discrete commutative ring \Phi with identity then there exists a structure of compact \Phi-algebra on A\otimes_\PhiB such that ( a\otimesb) ( a^'\otimesb^') = aa^'\otimesbb^' for a, a^' in A, b, b^' in B.

Date received: June 27, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caiv-31.