commutative ring

plural commutative rings

1. ring theoryalgebraA ring whose multiplicative operation is commutative.

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

   1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129

   2 more examples

   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.

   2002, Joseph J. Rotman, Advanced Modern Algebra, 2nd edition, American Mathematical Society, page 295

   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#92;inP, it follows that a#92;inP or b#92;inP; equivalently, P is a prime ideal if and only if the factor ring R#47;P is a domain.

   2004, K. R. Goodearl; R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge University Press, page 47