Direct decomposition of tensor products into subtensor products

A subtensor product of a family of modules is defined by using a subdirect product of the family of modules considered as sets. A tensor product of modules can be decomposed into a direct sum of subtensor products of the modules. Subtensor products of graded modules and graded algebras are also studied. As an application of these, a certain subtensor product of a family (not necessarily finite) of anticommutative algebras is shown to be a coproduct of this family in the category of unitary anticommutative algebras, and it can be imbedded as a direct summand into a tensor product of the family as modules. 1. Subtensor product of modules. Let (A),ei be a family of modules over a commutative ring R with unit, and U a subset of the cartesian product FIKI Ma as sets. For an R-module N, a mapping 9: U-+N will be called a multilinear mapping of U into N if for any (x)ei, (Y)ei and (Za)~ei in U such that x#=AyXfl+tzfl, where A and ,u are in R, for one fi EI, and x =yL=za for all = E I with cxof3, t(0a).sI) = A99((YcJcei) + P99((Zca)acXi) holds. A multilinear mapping of FLaei M. into N is a usual multilinear mapping; and a restriction mapping of this to U is an example of a multilinear mapping in our sense. By a similar construction to that of a tensor product of modules [1, Theorem 37, p. 87], the following theorem can be proved. THEOREM 1. Forany U ' lTLei M., there exist an R-module A and a multilinear mapping a of U into A such that for any R-module N andfor any multilinear mapping 9 of U into N there exists a unique linear mappingf of A into N such thatfo o= 9?. PROOF. Let F be a free R-module with U as its basis; and J be the submodule of F generated by the elements (X e)l a(y )aeI[-(Z zc)i,, where Presented to the Society, January 21, 1972; received by the editors September 28, 1971 and, in revised form, February 28, 1972. AMS (MOS) subject classifications (1969). Primary 1580; Secondary 1610, 1590.