commutative ring

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English[edit]

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Noun[edit]

commutative ring (plural commutative rings)

  1. (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.

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