##### Actions

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 , ). 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 , ). 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 , . 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 . Let \$ A \$ be a two-sided ideal of the ring \$ R \$; then

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

(see , ); 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) . Certain analogues of quasi-regular radicals exist in Jordan algebras (cf. Jordan algebra).

How to Cite This Entry: