commutative ring
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English[edit]
Noun[edit]
commutative ring (plural commutative rings)
 (algebra, ring theory) A ring whose multiplicative operation is commutative.
 1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129,
 Among commutative rings, the polynomial rings in a finite number of indeterminates enjoy important special properties and are frequently used in applications.
 2002, Joseph J. Rotman, Advanced Modern Algebra, American Mathematical Society, 2nd Edition, page 295,
 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.
 2004, K. R. Goodearl, R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cammbridge University Press, page 47,
 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 in a commutative ring is prime if, whenever we have two elements and of such that , it follows that or ; equivalently, is a prime ideal if and only if the factor ring is a domain.
 1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129,
Hyponyms[edit]
Translations[edit]
a ring whose multiplicative operation is commutative

