Noun
integral domain (plural integral domains)
(algebra, ring theory) Any nonzero commutative ring in which the product of nonzero elements is nonzero. [from 1911]
A ring
R
{\displaystyle R}
is an integral domain if and only if the polynomial ring
R
[
x
]
{\displaystyle R[x]}
is an integral domain.
For any integral domain there can be derived an associated field of fractions.
A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. Source: Internet
An integral domain is a Bézout domain if and only if any two elements in it have a gcd that is a linear combination of the two. Source: Internet
Field of fractions main The field of fractions K of an integral domain R is the set of fractions a/b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations. Source: Internet
A principal ideal domain is an integral domain in which every ideal is principal. Source: Internet
Equivalent conditions for a ring to be a UFD A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given below). Source: Internet
In particular, a commutative ring is an integral domain if and only if (0) is a prime ideal. Source: Internet