The Tensor Product

Abstract

Having considered bilinear maps, we now come to multilinear maps and basic theorems concerning their structure. There is a universal module representing multilinear maps, called the tensor product. We derive its basic properties, and postpone to Chapter XIX the special case of alternating products. The tensor product derives its name from the use made in differential geometry, when this product is applied to the tangent space or cotangent space of a manifold. The tensor product can be viewed also as providing a mechanism for “extending the base”; that is, passing from a module over a ring to a module over some algebra over the ring. This “extension” can also involve reduction modulo an ideal, because what matters is that we are given a ring homomorphism f: A→ B, and we pass from modules over A to modules over B. The homomorphism f can be of both types, an inclusion or a canonical map with B = A/J for some ideal J, or a composition of the two.