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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 [].
Another important example of a DVR is the ring of formal power series = [[]] in one variable over some field .The "unique" irreducible element is , the maximal ideal of is the principal ideal generated by , and the valuation assigns to each power series the index (i.e. degree) of the first non-zero coefficient.
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.
In abstract algebra, a valuation ring is an integral domain D such that for every non-zero element x of its field of fractions F, at least one of x or x −1 belongs to D.. Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F.
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 valuation ring R v is the set of a ∈ K with v(a) ≥ 0, the prime ideal m v is the set of a ∈ K with v(a) > 0 (it is in fact a maximal ideal of R v), the residue field k v = R v /m v, the place of K associated to v, the class of v under the equivalence defined below.
Given a linear operator T on a finite-dimensional vector space V, one can consider the vector space with operator as a module over the polynomial ring in one variable R = K[T], as in the structure theorem for finitely generated modules over a principal ideal domain. Then the spectrum of K[T] (as a ring) equals the spectrum of T (as an operator).
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}}.}