A fundamental theorem. If R is a given ring, we may sometimes be interested in constructing a ring S which contains R as a subring and which has some property not present in R. In this chapter, we shall consider some constructions of this type. However, before proceeding, we shall prove a fundamental theorem which states that it is often sufficient to construct a ring containing a subring isomorphic to R.

First we prove the following almost obvious

Lemma. If S is a ring, and T a set of elements in one-to-one correspondence with the elements of S, then addition and multiplication may be defined in T in such a way that T is a ring isomorphic to S.

Let us denote the given one-to-one correspondence by a ↔ a', where a is an element of S and a′ the corresponding element of T. To complete the proof it is only necessary to define addition and multiplication in T in the following natural way:

a′ + b′ = (a + b)′,

a′b′ = (ab)′,

and to observe that T is then a ring isomorphic to S.

Theorem 20. If R and S are rings with no elements in common, and S contains a subring S1 isomorphic to R, then there exists a ring T which is isomorphic to S and which contains R as a subring.