commutative ring

English

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, 2nd edition, American Mathematical Society, 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, Cambridge 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.

Hyponyms

• integral domain
  • unique factorization domain, Noetherian domain
    • principal ideal domain
      • Euclidean domain
        • field
• local ring
• commutative algebra
  • polynomial ring

Translations

a ring whose multiplicative operation is commutative

• Italian: anello commutativo m
• Japanese: 可換環 (ja) (かかんかん, kakankan)
• Romanian: inel comutativ n

Further reading

• Commutative algebra on Wikipedia.Wikipedia
• Commutative ring Encyclopedia of Mathematics