On graded Brown–McCoy radicals of graded rings

Abstract

We investigate the graded Brown-McCoy and the classical Brown-McCoy radical of a graded ring, which is the direct sum of a family of its additive subgroups indexed by a nonempty set, under the assumption that the product of homogeneous elements is again homogeneous. There are two kinds of the graded Brown-McCoy radical, the graded Brown-McCoy and the large graded Brown-McCoy radical of a graded ring. Several characterizations of the graded Brown-McCoy radical are given, and it is proved that the large graded Brown-McCoy radical of a graded ring is the largest homogeneous ideal contained in the classical Brown-McCoy radical of that ring.

On graded nil clean rings

“…We do not recall these notions here since we only need their properties. For more information on these and related radicals, one may also consult [11,12,13].…”

On the Jacobson and simple radicals of semigroups

“…Since homogeneous semigroups naturally arise from graded rings, it is easy to construct various examples of such semigroups. For examples of graded rings, see [11,12,21,23]. Clearly, the notion of a homogeneous semigroup generalizes the notion of a 0-band of semigroups and the notion of a band of semigroups.…”

On Graded Uj-Rings

“…The graded prime radical of R, denoted by P g (R), is defined as the intersection of all graded prime ideals of R (see [8]). For the graded versions of other classical radicals of rings, see [9,10,11]. This graded ring construction is too technical to present all of the details here.…”