×

On radicality of algebras graded by semigroups. (Russian) Zbl 0711.16009

Mat. Issled. 111, 82-96 (1989).
[For the entire collection see Zbl 0673.00010.]
Let \({\mathcal S}\) be a Kurosh-Amitsur radical class. \({\mathcal S}\) is said to be closed relative to a semigroup \(\Omega\) if every algebra \(R=\oplus_{\alpha \in \Omega}R_{\alpha}\) graded by \(\Omega\) satisfies \({\mathcal S}(R)=R\) provided that \({\mathcal S}(R_{\alpha})=R_{\alpha}\) for every component \(R_{\alpha}\) that is a subalgebra of R. The paper describes the classes of semigroups relative to which the radical classes in the sense of Jacobson, Levitzki, Baer, Andrunakievich-Ryabukhin and the hereditary idempotent radical, are closed. In particular, this is the class of all locally finite semigroups for the Jacobson and the Levitzki radicals, while a proper subclass of the class of locally finite semigroups in the case of the Baer radical. This is then applied to the graded Jacobson, Levitzki and Baer radicals of an algebra R (that is, the maximal homogeneous \({\mathcal S}\)-ideal of R contained in \({\mathcal S}(R))\) graded by a semigroup \(\Omega\). Namely, it is shown that the graded radical \({\mathcal S}_{\Omega}(R)\) is the largest among the homogeneous ideals I of R that satisfy \(I\cap R_{\epsilon}\leq {\mathcal S}(R_{\epsilon})\) for every \(\epsilon =\epsilon^ 2\in \Omega\), and for every \(\Omega\)-graded algebra R, if and only if \(\Omega\) belongs to the class of semigroups corresponding to \({\mathcal S}\).
Reviewer: J.Okniński

MSC:

16N60 Prime and semiprime associative rings
16W50 Graded rings and modules (associative rings and algebras)
20M99 Semigroups
16N20 Jacobson radical, quasimultiplication

Citations:

Zbl 0673.00010
Full Text: EuDML