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A radical class (also called radical property or just radical) is a class σ of rings possibly without multiplicative identities, such that: the homomorphic image 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
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). It is possible for a ring to have a unique maximal two-sided ideal and yet lack unique maximal one-sided ideals: for example ...
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 ...
Consider the ring of integers.. The radical of the ideal of integer multiples of is (the evens).; The radical of is .; The radical of is .; In general, the radical of is , where is the product of all distinct prime factors of , the largest square-free factor of (see Radical of an integer).
In abstract algebra, Jacobson's conjecture is an open problem in ring theory concerning the intersection of powers of the Jacobson radical of a Noetherian ring.. It has only been proven for special types of Noetherian rings, so far.