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Concept
Radical of a ring
Summary
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 , based on a suggestion of . In the next few years several other radicals were discovered, of which the most important example is the Jacobson radical. The general theory of radicals was defined independently by and .
In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have a multiplicative identity. In particular, every ideal in a ring is also a ring.
A radical class (also called radical property or just radical) is a class σ of rings possibly without identities, such that:
the homomorphic of a ring in σ is also in σ
every ring R contains an ideal S(R) in σ that contains every other ideal of R that is in σ
S(R/S(R)) = 0. The ideal S(R) is called the radical, or σ-radical, of R.
The study of such radicals is called torsion theory.
For any class δ of rings, there is a smallest radical class Lδ containing it, called the lower radical of δ. The operator L is called the lower radical operator.
A class of rings is called regular if every non-zero ideal of a ring in the class has a non-zero image in the class. For every regular class δ of rings, there is a largest radical class Uδ, called the upper radical of δ, having zero intersection with δ. The operator U is called the upper radical operator.
A class of rings is called hereditary if every ideal of a ring in the class also belongs to the class.
Jacobson radical
Let R be any ring, not necessarily commutative. 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.
J(R) is the maximal right (or left) quasi-regular right (resp. left) ideal of R.
Official source
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