Radical of A Ring - Definitions

Definitions

In the theory of radicals, rings are usually assumed to be associative, but need not be commutative and need not have an identity element. 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:

(1) the homomorphic image of a ring in σ is also in σ

(2) every ring R contains an ideal S(R) in σ which contains every other ideal in σ

(3) 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.