Meaning of integral domain | Babel Free

Definitions

Any nonzero commutative ring in which the product of nonzero elements is nonzero.

Equivalents

Examples

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

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

“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).”

“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.”

“#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.”

CEFR level