On complemented radicals
The authors obtain an explicit description of complemented radicals in the lattice of all radicals. It follows from their description that a radical \(\alpha\) is complemented in the lattice of all radicals if and only if either \(\alpha\) or its complement is a special radical determined by a finite family of matrix rings over finite fields. An ideal \(I\) of a ring \(A\) is essential if it has a nonzero intersection with any nonzero ideal of \(A\). Given an abstract class \(\alpha\) and a ring \(A\) with an ideal \(I\), then \(I\) is an \(\alpha\)-ideal if \(I\in\alpha\). Set \({\mathcal E}\alpha=\{A\mid A\) has an essential \(\alpha\)-ideal\}. The class \({\mathcal E}\alpha\) is called the essential cover of \(\alpha\). Denote the semisimple class of \(\alpha\) by \({\mathcal S}\alpha\). The authors study radicals \(\alpha\) satisfying one of the following conditions: (I) If \(A\) is a subdirect product of rings belonging to \({\mathcal E}\alpha\), then \(\alpha(A)\neq 0\). (II) If \(A\) is a subdirect product of rings belonging to \({\mathcal E}\alpha\), then \(\alpha(A)\) is an essential ideal of \(A\). The authors obtain four results about such rings: 1. If a radical \(\alpha\) satisfies (I), then either \(\alpha\) or \({\mathcal S}\alpha\) satisfies some proper polynomial identity. 2. If a radical \(\alpha\) satisfies (II), then either every finitely generated \(\alpha\)-semisimple ring is a direct sum of a finite number of matrix rings over finite fields, or for every finitely generated ring \(R\) there exists a central idempotent \(e\in R\) such that \(\alpha(R)=eR\). Furthermore, \(\alpha\) is complemented in the lattice of all radicals. 3. The authors give an explicit description of radicals \(\alpha\) with semisimple \({\mathcal E}\alpha\). They show that \({\mathcal E}\alpha\) is semisimple if and only if \(\alpha\) is hereditary and satisfies (I). 4. They obtain an explicit description of radicals \(\alpha\) such that every finitely generated \(\alpha\)-semisimple ring is a direct sum of simple rings.
scientific article; zbMATH DE number 1211666
scientific article