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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 [].
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).
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 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.
Consider a formal group F(X,Y) with coefficients in the ring of integers in a local field (for example Z p). Taking X and Y to be in the unique maximal ideal gives us a convergent power series and in this case we define F(X,Y) = X + F Y and we have a genuine group law. For example if F(X,Y)=X+Y, then this is the usual addition
More generally, if F is a local ring and n is a positive integer, then the quotient ring F[X]/(X n) is local with maximal ideal consisting of the classes of polynomials with constant term belonging to the maximal ideal of F, since one can use a geometric series to invert all other polynomials modulo X n.
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 ...
A related problem is the ideal membership problem, which consists in testing if a polynomial belongs to an ideal. For this problem also, a solution is provided by an upper bound on the degree of the g i. A general solution of the ideal membership problem provides an effective Nullstellensatz, at least for the weak form.