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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 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 radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by ...
In ring theory, a branch of mathematics, a radical of a ring is an ideal of "not-good" elements of the ring. The first example of a radical was the nilradical introduced by Köthe (1930), based on a suggestion of Wedderburn (1908). In the next few years several other radicals were discovered, of which the most important example is the Jacobson ...
radical 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.
A ring is called an SBI ring or Lift/rad ring if all idempotents of R lift modulo the Jacobson radical. A ring satisfies the ascending chain condition on right direct summands if and only if the ring satisfies the descending chain condition on left direct summands if and only if every set of pairwise orthogonal idempotents is finite.
A ring is called Jacobson semisimple (or J-semisimple or semiprimitive) if the intersection of the maximal left ideals is zero, that is, if the Jacobson radical is zero. Every ring that is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself.
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. If V is a maximal submodule of U, then U/V is simple.
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.