commutative ring

mathematics

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June 24, 2026

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In mathematics, a commutative ring is a set with two operations, addition and multiplication, that follow specific rules. Addition must be commutative (a + b = b + a) and associative, and there must be a zero element and additive inverses. Multiplication must be associative (a(bc) = (ab)c), and crucially for a commutative ring, multiplication must also be commutative (ab = ba). The set of integers with standard addition and multiplication is a fundamental example of a commutative ring. Modern algebra studies such structures, focusing on the properties of sets with operations rather than individual elements.12

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foundations of mathematics
One distinguished model or many models
…was the observation that every commutative ring may be viewed as a continuously variable local ring, as Lawvere would put it. In the same spirit, an amplified version of Gödel’s completeness theorem would say that every topos may be viewed as a continuously variable local topos, provided sufficiently many variables…
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…was the observation that every commutative ring may be viewed as a continuously variable local ring, as Lawvere would put it. In the same spirit, an amplified version of Gödel’s completeness theorem would say that every topos may be viewed as a continuously variable local topos, provided sufficiently many variables…
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ring
A commutative ring is a ring in which multiplication is commutative—that is, in which ab = ba for any a, b.
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A commutative ring is a ring in which multiplication is commutative—that is, in which ab = ba for any a, b.
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modern algebra
Structural axioms
…8) is known as a commutative ring. When axioms 1–9 hold and there are no proper divisors of zero (i.e., whenever ab = 0 either a = 0 or b = 0), a set is called an integral domain. For example, the set of integers {…, −2, −1, 0, 1,…
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…8) is known as a commutative ring. When axioms 1–9 hold and there are no proper divisors of zero (i.e., whenever ab = 0 either a = 0 or b = 0), a set is called an integral domain. For example, the set of integers {…, −2, −1, 0, 1,…
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