An example of how to select ideals to obtain quotient rings with desirable properties

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

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

The example under consideration is stated as follows. Let A be a commutative ring with unity. An ideal J of A is prime if and only if A/J 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  J of a commutative ring A is prime if \forall a,b\in A with ab\in J it follows that a\in J or b \in J.

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

Think first how the lack of divisors of zero is stated for A/J. The product of two cosets J+a and J+b in A/J is (J+a)(J+b)=J+ab. Moreover, the zero element of A/J is the coset J=J+0, so to say that A/J has no divisors of zero implies that J+ab=J\Rightarrow J+a=J or J+b=J.

Recall also that for any x\in A, x\in J \Leftrightarrow J+x=J. It then becomes obvious that the property J+ab=J\Rightarrow J+a=J or J+b=J in A/J corresponds to the property ab\in J\Rightarrow a\in J or b\in J, which is the defining property of a prime ideal J.

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


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