# commutative ring

## English

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### Noun

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 ${\displaystyle P}$ in a commutative ring ${\displaystyle R}$ is prime if, whenever we have two elements ${\displaystyle a}$ and ${\displaystyle b}$ of ${\displaystyle R}$ such that ${\displaystyle ab\in P}$, it follows that ${\displaystyle a\in P}$ or ${\displaystyle b\in P}$; equivalently, ${\displaystyle P}$ is a prime ideal if and only if the factor ring ${\displaystyle R/P}$ is a domain.