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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

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)
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