Tensor Products of Modules

The notion of a tensor product of topological groups and modules is important in theory of topological groups, algebraic number theory. The tensor product of compact zero-dimensional modules over a pseudocompact algebra was introduced in [B] and for the commutative case in [GD], [L]. The notion of a tensor product of abelian groups was introduced in [H]. The tensor product of modules over commutative topological rings was given in [AU]. We will construct in this note the tensor product of a right compact R-module AR and a left compact R-module RB over a topological ring R with identity. Some properties of tensor products are given. Notation ω stands for the set of all natural numbers. If A is a locally compact group, K a compact subset of it and ɛ>0, then T(K,ɛ) := {α∈A* : α(K) ⊆ φ (Oɛ)}, where φ is the canonical homomorphism of R on R/Z = T. If m, n are natural numbers then [m, n] stands for the set of all natural numbers x such that m ≤ x ≤ n. Let R be a topological ring with identity and AR, RB compact unitary right and left R-modules, respectively. A continuous function β : A×B → C, where C is a compact abelian group, is said to be R-balanced if it is linear on each variable, i.e., β(a1+a2,b) = β(a1,b) + β(a2,b) and β(a,b1+b2) = β(a,b1) + β(a,b2) for each a, a1, a2∈A, b, b1, b2∈ B, and β(ar, b) = β(a, rb) for each a∈A, b∈B, r∈R. A pair (C, π) where C is a compact abelian group and π : A×B → C is a Rbalanced map is called a tensor product of AR and RB provided for each compact abelian group D and each R-balanced mapping α : A×B → D there exists a unique continuous homomorphism α : C → D such that the following diagram commutes,