commutative ring
Noun
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Noun
commutative ring (plural commutative rings)
- (algebra, ring theory) A ring#Etymology_3|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 P in a commutative ring R is prime if, whenever we have two elements a and b of R such that ab\in P, it follows that a\in P or b\in P; equivalently, P is a prime ideal if and only if the factor ring R/P is a domain.
- 1960, Oscar Zariski, Pierre Samuel, Commutative Algebra II, Springer, page 129 ↗,
- Italian: anello commutativo
This text is extracted from the Wiktionary and it is available under the CC BY-SA 3.0 license | Terms and conditions | Privacy policy 0.004