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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
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}}.}
A prime ideal has height zero if and only if it is a minimal prime ideal. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal. In a Noetherian ring, every prime ideal
If x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax + by, etc. Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains.
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.
A subset I of a partially ordered set (,) is an ideal, if the following conditions hold: [1] [2] I is non-empty, for every x in I and y in P, y ≤ x implies that y is in I (I is a lower set), for every x, y in I, there is some element z in I, such that x ≤ z and y ≤ z (I is a directed set).