General remarks

Our main objective is to discuss the important algebras that can be derived from a module, and so in this, the last of the introductory chapters, we shall study those general aspects of the theory of Algebras that are relevant to our goal. At the outset it needs to be stressed that we shall be concerned solely with algebras that are associative and possess an identity element. Accordingly, from now on, the term algebra will only be used in this restricted sense.

As before R and S denote commutative rings, each with an identity element, and a ring-homomorphism between them is required to take identity element into identity element. Later, when we come to consider homomorphisms of algebras, these too will be required to preserve identity elements.

Finally when dealing with tensor products of modules we shall sometimes omit from the symbol ⊗ the suffix which indicates the underlying ring. In this chapter whenever the suffix is left out the ring in question is always R.

Basic definitions

Let A be a (not necessarily commutative) ring with an identity element 1A, and at the same time let it be a module over the commutative ring R. We suppose that the sum of two elements of A is the same whether we use the ring or the module structure.