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of a ring

The ideal of an associative ring (cf. Associative rings and algebras) which satisfies the following two requirements: 1) is the largest quasi-regular ideal in (a ring is called quasi-regular if the equation is solvable for any of its elements ); and 2) the quotient ring contains no non-zero quasi-regular ideals. The radical was introduced and studied in detail in 1945 by N. Jacobson [1].

The Jacobson radical always exists and may be characterized in very many ways: is the intersection of the kernels of all irreducible representations of the ring ; it is the intersection of all modular maximal right ideals (cf. Modular ideal); it is the intersection of all modular maximal left ideals; it contains all quasi-regular one-sided ideals; it contains all one-sided nil ideals; etc. If is an ideal of , then . If is the ring of all matrices of order over , then

If the following -composition is introduced on the associative ring :

then the radical in the semi-group will be a subgroup with respect to the composition .

There are no non-zero irreducible finitely-generated modules over a quasi-regular associative ring (i.e. an associative ring coinciding with its own Jacobson radical), but there exist simple associative quasi-regular rings. The Jacobson radical of the associative ring is zero if and only if is a subdirect sum of primitive rings (cf. Primitive ring).

References

 [1] N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)