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A ring R is called a Jacobson ring if the nilradical and Jacobson radical of R/P coincide for all prime ideals P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.
If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.
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 ...
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent. [1] [2]The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil.
There are, however, analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed. The concept of the Jacobson radical of a ring; that is, the intersection of all right (left) annihilators of simple right (left) modules over a ring, is one example. The fact that the Jacobson ...
1. The Jacobson radical of a ring. 2. The nilradical of a ring. 3. A radical of an element x of a ring is an element such that some positive power is x. 4. The radical of an ideal is the ideal of radicals of its elements. 5. The radical of a submodule M of a module N is the ideal of elements x such that some power of x maps N into M. 6.
Jill Jacobson, a star of film and TV known for her work in Star Trek: The Next Generation and the soap operas Falcon Crest and Days of Our Lives, has died.She was 70 years old. Jacobson's friend ...
Any finitely generated algebra over a Jacobson ring is a Jacobson ring. In particular, any finitely generated algebra over a field or the integers, such as the coordinate ring of any affine algebraic set, is a Jacobson ring. A local ring has exactly one maximal ideal, so it is a Jacobson ring exactly when that maximal ideal is the only prime ideal.