On graded Brown-McCoy radicals of graded rings (Q2816909)

All rings in this paper are assumed to be associative. Let \(R\) be a ring and \(S\) a set with a partial binary operation. Let \(\left\{ R_{s}\right\} _{s\in S}\) be a family of additive subgroups of \(R\). The set \(R=\oplus _{s\in S}R_{s}\) is said to be an \(S\)-graded ring inducing \(S\) if the following two conditions hold: (i) \(R_{s}R_{t}\subset R_{st}\) whenever \(st\) is defined and (ii) \(R_{s}R_{t}\neq 0\) implies that the product \(st\) is defined. In this paper, an \(S\)-graded ring inducing \(S\) is simply called a graded ring. If \(R=\oplus _{s\in S}R_{s}\) is a graded ring, the set \(A=\cup _{s\in S}R_{s}\) is called the homogeneous part of \(R\) and elements of \(A\) are called homogeneous elements of \(R\). Moreover, the set \(A\) with induced partial addition and induced multiplication from \(R\) is called an anneid.NEWLINENEWLINEAs studying graded rings is equivalent to studying their corresponding anneids, in this paper the author studies anneids all of which are assumed to be associative. In particular, the author introduces and investigates the graded Brown-McCoy radical of an anneid. Two notions of this radical are given: the graded Brown-McCoy and the large graded Brown-McCoy radical of an anneid. They are too technical to be included in this review. The author gives several characterizations of the graded Brown-McCoy radical of an anneid and shows that the large graded Brown-McCoy radical of an anneid is the homogeneous part of the largest homogeneous ideal contained in the classical Brown-McCoy radical of the corresponding graded ring. Results presented in this paper are generalizations of results which hold for usual group-graded rings.