Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications.

As usual, it is simpler to begin by looking at a more general setting—in this case, commutative rings—before getting involved with polynomial rings. It turns out that the nature of the ideals in a commutative ring is important: for example, we have already seen that gcd's exist in PIDs, while this may not be true in other commutative rings.

In trying to understand the ideal theory of a commutative ring, one quickly sees that it is important to first understand the prime ideals. We recall that a proper ideal P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab∈P, it follows that a∈P or b∈P; equivalently, P is a prime ideal if and only if the factor ring R/P is a domain.

Commutative ring

Meaning

Noun