integral domain

plural integral domains

1. ring theoryalgebraAny nonzero commutative ring in which the product of nonzero elements is nonzero.First known use 1911

   For any integral domain there can be derived an associated field of fractions.

   4 more examples

   A ring R is an integral domain if and only if the polynomial ring R#91;x#93; is an integral domain.

   For integral domains, we will use a⁻¹ to designate the multiplicative inverse of a (if it has one; since not all elements need have inverses, this notation can be used only where it can be shown that an inverse exists).

   1990, Barbara H. Partee, Alice ter Meulen, Robert E. Wall, Mathematical Methods in Linguistics, Kluwer Academic Publishers, page 266

   An integral domain is said to have strong pseudo-Dedekind factorization if each proper ideal can be factored as the product of an invertible ideal (possibly equal to the ring) and a finite product of pairwise comaximal prime ideals with at least one prime in the product.

   2013, Marco Fontana, Evan Houston, Thomas Lucas, Factoring Ideals in Integral Domains, Springer, page 95

   #91;#92;mathbb#123;Z#125;#59;#43;,#92;cdot#93;, #91;#92;mathbb#123;Z#125;#95;p,#43;#95;p,#92;times#95;p#93; with p a prime, #91;#92;mathbb#123;Q#125;#59;#43;,#92;cdot#93;, #91;#92;mathbb#123;R#125;#59;#43;,#92;cdot#93;, and #91;#92;mathbb#123;C#125;#59;#43;,#92;cdot#93; are all integral domains. The key example of an infinite integral domain is #91;#92;mathbb#123;Z#125;#59;#43;,#92;cdot#93;. In fact, it is from #92;mathbb#123;Z#125; that the term integral domain is derived. Our main example of a finite integral domain is #91;#92;mathbb#123;Z#125;#95;p,#43;#95;p,#92;times#95;p#93;, when p is prime.

   2017, Ken Levasseur, Al Doerr, Applied Discrete Structures: Part 2 - Applied Algebra, Lulu.com, page 171

   Synonyms: entire ring