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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 ...
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.
R is a Jacobson ring; Every prime ideal of R is an intersection of maximal ideals. Every radical ideal is an intersection of maximal ideals. Every Goldman ideal is maximal. Every quotient ring of R by a prime ideal has a zero Jacobson radical. In every quotient ring, the nilradical is equal to the Jacobson radical.
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.
The following simple but important fact (Nakayama's lemma) is built-in to the definition of a Jacobson radical: if M is a module such that = , then M does not admit a maximal submodule, since if there is a maximal submodule , (/) = and so = , a contradiction.