##### Actions

2010 Mathematics Subject Classification: Primary: 16N20 [MSN][ZBL]

of a ring $A$

The ideal $J(A)$ of an associative ring (cf. Associative rings and algebras) $A$ which satisfies the following two requirements: 1) $J(A)$ is the largest quasi-regular ideal in $A$ (a ring $R$ is called quasi-regular if the equation $a+x+ax=0$ is solvable for any of its elements $a$; cf. Quasi-regular ring); and 2) the quotient ring $\overline A=A/J(A)$ contains no non-zero quasi-regular ideals. The radical was introduced and studied in detail in 1945 by N. Jacobson .

The Jacobson radical always exists and may be characterized in very many ways: $J(A)$ is the intersection of the kernels of all irreducible representations of the ring $A$; 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 $I$ is an ideal of $A$, then $J(I)=I\cap J(A)$. If $A_n$ is the ring of all matrices of order $n$ over $A$, then

$$J(A_n)=(J(A))_n.$$

If the following $\circ$-composition is introduced on the associative ring $A$:

$$a\circ b=a+b+ab,$$

then the radical $J(A)$ in the semi-group $\langle A,\circ\rangle$ will be a subgroup with respect to the composition $\circ$.

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 $A$ is zero if and only if $A$ is a subdirect sum of primitive rings.

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