Quasi-regular radical

of a ring

The largest quasi-regular ideal of the given ring. An ideal $ A $ of a ring $ R $ is called quasi-regular if $ A $ is a quasi-regular ring. There exists in every alternative (in particular, associative) ring a quasi-regular radical; it coincides with the sum of all right (left) quasi-regular ideals (see [1], [10]). The quasi-regular radical of an associative ring is also called the Jacobson radical.

The quasi-regular radical $ J ( R) $ of an arbitrary alternative ring $ R $ is equal to the intersection of all maximal modular right (left) ideals of $ R $; $ J ( R) $ is also equal to the intersection of the kernels of all irreducible right (left) representations of $ R $( see [1], [5]–[8]). A ring $ R $ is called $ J $- semi-simple if $ J ( R) = 0 $. The quotient ring $ R / J ( R) $ is always semi-simple. Every semi-simple ring is isomorphic to a subdirect sum of primitive rings [1], [8]. If $ R $ satisfies the minimum condition for right (left) ideals, then the radical $ J ( R) $ is nilpotent and the quotient ring $ R / J ( R) $ is isomorphic to a finite direct sum of complete matrix rings over skew-fields and Cayley–Dickson algebras (the latter terms are absent in the associative case), see [1]–[3]. Let $ A $ be a two-sided ideal of the ring $ R $; then

$$ J ( A) = A \cap J ( R) $$

(see [1], [4]); if $ R $ is an associative ring and $ R _ {n} $ is the ring of matrices of order $ n $ over $ R $, then

$$ J ( R _ {n} ) = [ J ( R) ] _ {n} . $$

If $ R $ is an associative algebra over a field $ F $ and the cardinality of $ F $ is greater than the dimension of $ R $ over $ F $ or if $ R $ is algebraic over $ F $, then $ J ( R) $ is a nil ideal. A quasi-regular radical of a finitely-generated alternative ring satisfying an essential identity relation is the same as a lower nil radical (see Radical of rings and algebras) [6]. Certain analogues of quasi-regular radicals exist in Jordan algebras (cf. Jordan algebra).

References

[1]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[2]	K.A. Zhevlakov, "On radical ideals of an alternative ring" Algebra i Logika , 4 : 4 (1965) pp. 87–102 (In Russian)
[3]	K.A. Zhevlakov, "Alternative Artinian rings" Algebra i Logika , 5 : 3 (1966) pp. 11–36 (In Russian)
[4]	K.A. Zhevlakov, "On the Kleinfeld and Smiley radicals of alternative rings" Algebra and Logic , 8 : 2 (1969) pp. 100–102 Algebra i Logika , 8 : 2 (1969) pp. 176–180
[5]	K.A. Zhevlakov, "Coincidence of Kleinfeld and Smiley radicals in alternative rings" Algebra and Logic , 8 : 3 (1969) pp. 175–181 Algebra i Logika , 8 : 3 (1969) pp. 309–319
[6]	K.A. Zhevlakov, "Quasiregular ideals in finitely generated alternative rings" Algebra and Logic , 11 : 2 (1972) pp. 94–100 Algebra i Logika , 11 : 2 (1972) pp. 140–1161
[7]	A.M. Slin'ko, I.P. Shestakov, "Right representation of algebras" Algebra and Logic , 13 : 5 (1973) pp. 312–333 Algebra i Logika , 13 : 5 (1974) pp. 544–588
[8]	E. Kleinfeld, "Primitive alternative rings and semi-simplicity" Amer. J. Math. , 77 (1955) pp. 725–730
[9]	K. McCrimmon, "The radical of a Jordan algebra" Proc. Nat. Acad. Sci. USA , 62 (1969) pp. 671–678
[10]	M.F. Smiley, "The radical of an alternative ring" Ann. of Math. , 49 : 3 (1948) pp. 702–709

Comments

The radical of a row-finite infinite matrix ring is determined by annihilating sequences of ideals [a1]. The radical of a polynomial ring $ R [ x] $ is $ N [ x] $ for some nil ideal $ N $ in $ R $( the determination of $ N $ is an open problem).

References

[a1]	N.E. Sexauer, J.E. Warnock, "The radical of the row-finite matrices over an arbitrary ring" Trans. Amer. Math. Soc. , 39 (1969) pp. 281–295
[a2]	L.H. Rowen, "Ring theory" , I, II , Acad. Press (1988)