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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 [].
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.
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.
An integral domain where every finitely generated ideal is principal (that is, a Bézout domain) satisfies (ACCP) if and only if it is a principal ideal domain. [4] The ring Z+XQ[X] of all rational polynomials with integral constant term is an example of an integral domain (actually a GCD domain) that does not satisfy (ACCP), for the chain of ...
For example, x, y(1-x), z(1-x) is a regular sequence in the polynomial ring C[x, y, z], while y(1-x), z(1-x), x is not a regular sequence. But if R is a Noetherian local ring and the elements r i are in the maximal ideal, or if R is a graded ring and the r i are homogeneous of positive degree, then any permutation of a regular sequence is a ...
Thus, points in n-space, thought of as the max spec of = [, …,], correspond precisely to 1-dimensional representations of R, while finite sets of points correspond to finite-dimensional representations (which are reducible, corresponding geometrically to being a union, and algebraically to not being a prime ideal). The non-maximal ideals then ...
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.