Chapter I - THE PRIMARY DECOMPOSITION

Summary

A convention. Now that we have given a formal definition of a ring, we can begin the systematic development of our subject. The rings that we shall consider will all be commutative, and they will all have a unit element. It is therefore convenient to use the word ‘ring’ in a more restricted sense than is customary in modern algebra, and for this reason we lay down the following convention: From now on ‘ring’ will always mean a commutative ring with a unit element. The zero element and the unit element of a ring R will be denoted by 0 and 1 respectively, or, if we are concerned with several rings at the same time, by 0_R and I_R.

Ideals and their calculus. Let R be a ring (commutative and with a unit element), and let a be a non-empty subset of R, then a is called an ideal of R in all cases where the following two conditions are satisfied:

1. Whenever a₁ and a₂ belong to a, then a₁ ± a2 both belong to a.

2. If a ∈ a, then ra ∈ a for all r ∈ R.

A trivial example of an ideal is obtained by taking a to be the whole ring.