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There is an analogous list for one-sided ideals, for which only the right-hand versions will be given. For a right ideal A of a ring R, the following conditions are equivalent to A being a maximal right ideal of R: There exists no other proper right ideal B of R so that A ⊊ B. For any right ideal B with A ⊆ B, either B = A or B = R.
For a general ring with unity R, the Jacobson radical J(R) is defined as the ideal of all elements r ∈ R such that rM = 0 whenever M is a simple R-module.That is, = {=}. This is equivalent to the definition in the commutative case for a commutative ring R because the simple modules over a commutative ring are of the form R / for some maximal ideal of R, and the annihilators of R / in R are ...
For noncommutative rings, the analogues for maximal left ideals and maximal right ideals also hold. For pseudo-rings, the theorem holds for regular ideals. An apparently slightly stronger (but equivalent) result, which can be proved in a similar fashion, is as follows: Let R be a ring, and let I be a proper ideal of R.
J(R) is the intersection of all the right (or left) primitive ideals of R. J(R) is the maximal right (or left) quasi-regular right (resp. left) ideal of R. As with the nilradical, we can extend this definition to arbitrary two-sided ideals I by defining J(I) to be the preimage of J(R/I) under the projection map R → R/I.
(DD2) is Noetherian, and the localization at each maximal ideal is a discrete valuation ring. (DD3) Every nonzero fractional ideal of R {\displaystyle R} is invertible. (DD4) R {\displaystyle R} is an integrally closed , Noetherian domain with Krull dimension one (that is, every nonzero prime ideal is maximal).
The intersection of all maximal right ideals which are modular is the Jacobson radical. [8] Examples. In the non-unital ring of even integers, (6) is regular (=) while (4) is not. Let M be a simple right A-module. If x is a nonzero element in M, then the annihilator of x is a regular maximal right ideal in A.
Americanism, also referred to as American patriotism, is a set of patriotic values which aim to create a collective American identity for the United States that can be defined as "an articulation of the nation's rightful place in the world, a set of traditions, a political language, and a cultural style imbued with political meaning". [1]
The converse is also true: if a prime ideal has height n, then it is a minimal prime ideal over an ideal generated by n elements. [ 1 ] The principal ideal theorem and the generalization, the height theorem, both follow from the fundamental theorem of dimension theory in commutative algebra (see also below for the direct proofs).