The fundamental homomorphism theorem for rings employs the kernel of a homomorphism to construct a quotient ring isomorphic to ring . Conversely, the homomorphic image of a ring is useful for factoring out unwanted features of to obtain a quotient ring with the desirable properties. Selecting an appropriate ideal is the key for acquiring the desired quotient ring .

In some cases, picking a suitable ideal can be done methodically by tracing the desired property of back to an analogous property of . To demonstrate the process of selecting , the simple example of obtaining a quotient ring that doesn’t contain any divisors of zero will be considered.

The example under consideration is stated as follows. Let be a commutative ring with unity. An ideal of is prime if and only if is an integral domain.

Briefly recall the associated definitions. An integral domain is a commutative ring with unity that doesn’t have any divisors of zero. An ideal of a commutative ring is prime if with it follows that or .

It is straightforward to prove the equivalence of this example. To prove that the integral domain has no divisors, it suffices to observe that . Emphasis will be shifted towards understanding why one would think of using a prime ideal to retain the cancellation property in , i.e. to ensure that there are no divisors of zero in .

Think first how the lack of divisors of zero is stated for . The product of two cosets and in is . Moreover, the zero element of is the coset , so to say that has no divisors of zero implies that or .

Recall also that for any , . It then becomes obvious that the property or in corresponds to the property or , which is the defining property of a prime ideal .

So it appears that in some cases it is easy to choose an ideal whose properties are conveyed to a quotient ring .