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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 ...
The Jacobson radical of R is the intersection of the annihilators of all simple right R-modules. There are several equivalent characterizations of the Jacobson radical, such as: J(R) is the intersection of the regular maximal right (or left) ideals of R. J(R) is the intersection of all the right (or left) primitive ideals of R.
For a ring R with Jacobson radical J, the nonnegative powers are defined by using the product of ideals.. Jacobson's conjecture: In a right-and-left Noetherian ring, = {}. In other words: "The only element of a Noetherian ring in all powers of J is 0."
The resulting theorem is sometimes known as the Jacobson–Azumaya theorem. [13] Let J(R) be the Jacobson radical of R. If U is a right module over a ring, R, and I is a right ideal in R, then define U·I to be the set of all (finite) sums of elements of the form u·i, where · is simply the action of R on U. Necessarily, U·I is a submodule of U.
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. Every finitely generated algebra over R that is a field is finitely generated as an R-module. (Zariski's lemma) Every prime ideal P of R such that R/P has an element x with (R/P)[x −1] a field is a ...
If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J(R).
In mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. (Lam 2001, p. §20)(Mikhalev & Pilz 2002, p. C.7) The above definition is satisfied if R has a finite number of maximal right ideals (and
Furthermore, every simple (left) R-module is isomorphic to a minimal left ideal of R, that is, R is a left Kasch ring. Semisimple rings are both Artinian and Noetherian. From the above properties, a ring is semisimple if and only if it is Artinian and its Jacobson radical is zero.