Search results
Results From The WOW.Com Content Network
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.
Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number p and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and p is replaced by any maximal ideal (indeed, the maximal ideals of have the form , where p is a prime number).
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 Boolean algebras, the terms prime ideal and maximal ideal coincide, as do the terms prime filter and maximal filter. There is another interesting notion of maximality of ideals: Consider an ideal I and a filter F such that I is disjoint from F. We are interested in an ideal M that is maximal among all ideals that contain I and are disjoint ...
Let m be the ideal (x), and let n be the ideal generated by x–1 and all the elements z i. These are both maximal ideals of R, with residue fields isomorphic to k. The local ring R m is a regular local ring of dimension 1 (the proof of this uses the fact that z and x are algebraically independent) and the local ring R n is a regular Noetherian ...
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.
Z-modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups.These are the cyclic groups of prime order.. If I is a right ideal of R, then I is simple as a right module if and only if I is a minimal non-zero right ideal: If M is a non-zero proper submodule of I, then it is also a right ideal, so I is not minimal.
Let M be an R-module generated by n elements, and φ: M → M an R-linear map. If there is an ideal I of R such that φ(M) ⊂ IM, then there is a monic polynomial = + + + with p k ∈ I k, such that = as an endomorphism of M.