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If R is a unital commutative ring with an ideal m, then k = R/m is a field if and only if m is a maximal ideal. In that case, R/m is known as the residue field. This fact can fail in non-unital rings. For example, is a maximal ideal in , but / is not a field. If L is a maximal left ideal, then R/L is a simple left R-module.
The maximal ideal of () is the principal ideal generated by 2, i.e. (), and the "unique" irreducible element (up to units) is 2 (this is also known as a uniformizing parameter). Note that Z ( 2 ) {\displaystyle \mathbb {Z} _{(2)}} is the localization of the Dedekind domain Z {\displaystyle \mathbb {Z} } at the prime ideal generated by 2.
Frequently, is a local ring and is then its unique maximal ideal. In abstract algebra, the splitting field of a polynomial is constructed using residue fields. Residue fields also applied in algebraic geometry , where to every point x {\displaystyle x} of a scheme X {\displaystyle X} one associates its residue field k ( x ) {\displaystyle k(x ...
Maximal ideal: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of J. The factor ring of a maximal ideal is a simple ring in general and is a field for commutative rings. [12] Minimal ideal: A nonzero ideal is called minimal if it contains no other nonzero ideal.
m is a minimal prime over (x 1, ..., x d). The radical of (x 1, ..., x d) is m. Some power of m is contained in (x 1, ..., x d). (x 1, ..., x d) is m-primary. Every local Noetherian ring admits a system of parameters. [1] It is not possible for fewer than d elements to generate an ideal whose radical is m because then the dimension of R would ...
An integral domain where every finitely generated ideal is principal (that is, a Bézout domain) satisfies (ACCP) if and only if it is a principal ideal domain. [4] The ring Z+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the chain of ...