Search results
Results from the WOW.Com Content Network
A commutative ring R is an integral domain if and only if the ideal (0) of R is a prime ideal. If R is a commutative ring and P is an ideal in R, then the quotient ring R/P is an integral domain if and only if P is a prime ideal. Let R be an integral domain. Then the polynomial rings over R (in any number of
In algebra, a domain is a nonzero ring in which ab = 0 implies a = 0 or b = 0. [1] (Sometimes such a ring is said to "have the zero-product property".) Equivalently, a domain is a ring in which 0 is the only left zero divisor (or equivalently, the only right zero divisor). A commutative domain is called an integral domain.
A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given at the end). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case, it is in fact a principal ideal domain. In general, for an integral domain A, the following conditions are equivalent: A is a UFD.
It follows immediately that, if K is an integral domain, then so is K[X]. [13] It follows also that, if K is an integral domain, a polynomial is a unit (that is, it has a multiplicative inverse) if and only if it is constant and is a unit in K. Two polynomials are associated if either one is the product of the other by a unit.
Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers.
For a noetherian local domain A of dimension one, the following are equivalent. A is integrally closed. The maximal ideal of A is principal. A is a discrete valuation ring (equivalently A is Dedekind.) A is a regular local ring. Let A be a noetherian integral domain.
In principal ideal domains a near converse holds: every nonzero prime ideal is maximal. All principal ideal domains are integrally closed. The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain. Let A be an integral domain, the following are equivalent. A is a PID.
In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A. [1]If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).