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If F is a field, then the only maximal ideal is {0}. In the ring Z of integers, the maximal ideals are the principal ideals generated by a prime number. More generally, all nonzero prime ideals are maximal in a principal ideal domain. The ideal (,) is a maximal ideal in ring [].
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.
In any ring R, a maximal ideal is an ideal M that is maximal in the set of all proper ideals of R, i.e. M is contained in exactly two ideals of R, namely M itself and the whole ring R. Every maximal ideal is in fact prime. In a principal ideal domain every nonzero prime ideal is maximal
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.
The completion of a Noetherian local ring with respect to the unique maximal ideal is a Noetherian local ring. [ 3 ] The completion is a functorial operation: a continuous map f : R → S of topological rings gives rise to a map of their completions, f ^ : R ^ → S ^ . {\displaystyle {\widehat {f}}:{\widehat {R}}\to {\widehat {S}}.}
In all unital rings, maximal ideals are prime. 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.
For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...
Still more generally, if A is a regular local ring, then the formal power series ring A[[x]] is regular local. If Z is the ring of integers and X is an indeterminate, the ring Z[X] (2, X) (i.e. the ring Z[X] localized in the prime ideal (2, X) ) is an example of a 2-dimensional regular local ring which does not contain a field.