Graded varieties of graded rings (Q1804734)

\(G\)-graded rings with an identity are considered where \(G\) is a finite group. First the concept of a graded variety is introduced and the graded version of Birkhoff's Theorem is proved. A proper subclass \({\mathcal V}\) of all \(G\)-graded rings is a graded radical graded semisimple class if and only if \({\mathcal V} \subseteq {\mathcal D}^g = \{R : R \) is a graded ring, \([x]= [x]^2\), \(\forall x \in h(R)\}\) where \(h(R)\) denotes the homogeneous elements of \(R\). A variety \({\mathcal V}^g(P,N)\) of graded rings is constructed via a finite set \(P\) of primes and of positive integers \(N\) which is a graded radical graded semisimple class, and every graded radical graded semisimple class is contained in some \({\mathcal V}^g (P,N)\). A graded radical and graded semisimple class \(\mathcal K\) equals such a \({\mathcal V}^g (P,N)\) if and only if \(\mathcal K\) is determined by a finite set of finite graded division rings. Assuming that \(G\) is cyclic, the defining equations of \(\mathcal K\) are given explicitly, in particular, for the class \({\mathcal K}_m = \{R : R\) is a graded ring, \(x^m = x\), \(\forall x \in h(R)\}\) also the graded division rings defining \({\mathcal K}_m\) are determined for \(m \mid O(G)\). The proofs involve number theoretic arguments. Also examples are given illustrating unexpected situations.