Meaning of commutative ring | Babel Free

Definitions

A ring whose multiplicative operation is commutative.

Equivalents

Examples

“Among commutative rings, the polynomial rings in a finite number⟳ of indeterminates enjoy⟳ important special properties and are frequently used in applications.”

“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.”

“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#92;inP, it follows that a#92;inP or b#92;inP; equivalently, P is a prime ideal if and only if the factor ring⟳ R#47;P is a domain.”

CEFR level