Graded radical graded semisimple classes (Q1804759)

In 1970, P. N. Stewart gave a characterization of radical semisimple classes of associative rings. He showed that if \(C\) is a proper subclass of all associative rings, then \(C\) is a radical semisimple class if and only if there is a strongly hereditary finite set \(C(F)\) of finite fields such that \(R \in C\) if and only if \(R\) is isomorphic to a subdirect sum of fields in \(C(F)\). In 1992, H. Fang and P. N. Stewart gave some examples of graded radical graded semisimple classes. They mention that it remains an open question how to characterize such classes. In this paper, the author answers their question. Let \(G\) be a multiplicative group with identity element \(e\). A \(G\)-graded ring \(R\) is a ring together with a direct sum decomposition \(R = \oplus_{g \in G} R_ g\), where \(R_ g\), \(g \in G\), is an additive subgroup of \(R\) such that \(R_ g R_ h \subseteq R_{gh}\) for all \(g, h \in G\). The abelian subgroup \(R_ g\) is called the homogeneous \(g\)- component of \(R\). Let \(h(R)\) denote the set of all homogeneous elements of \(R\), so \(h(R) = \bigcup_{g \in G} R_ g\). Throughout the paper the author considers graded rings, graded by a finite group \(G\) of order \(n\). If \(x \in R\), let \([x]\) denote the subring of \(R\) generated by \(x\). A \(G\)-graded ring \(R\) is a \(D^ g\)-ring if for each \(x \in h(R)\) we have \([x] = [x]^ 2\). The author shows that the class of all \(D^ g\)-rings is a graded radical class. The author then shows that if a graded radical graded semisimple class \(K\) does not consist of all the graded rings, then \(K\) is a \(D^ g\)-ring. A class \(K\) of graded rings is called graded strongly hereditary if every homogeneous subring of a ring in \(K\) is also in \(K\). The author obtains characterizations of a graded radical semisimple class \(K\) in terms of a graded strongly hereditary finite set of finite graded division rings. In the final section of the paper, the author gives graded versions of some results of Andrunakievich. He also gives another characterization of a graded radical graded semisimple class in terms of a graded special radical and its dual graded radical.