On graded special radicals of graded rings

Abstract

In this paper, a graded ring is a 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. We study graded special radicals and special radicals of graded rings, but which contain the corresponding Jacobson radicals. There are two versions of this graded radical, which we name the graded over-Jacobson and the large graded over-Jacobson radical. We establish several characterizations of the graded over-Jacobson radical of a graded ring and also prove that the largest homogeneous ideal contained in the corresponding classical radical of a graded ring coincides with the large graded over-Jacobson radical of that ring.

On graded nil clean rings

“…Since R is a graded local ring, we have that R/J(R) is a graded division ring. If [13]). It is easily seen that f is well defined and that it is a surjective homomorphism of anneids.…”

“…∈ R e or x ∈ J(R), and where x + J(R) ∩ H R ∈ H R/J(R) (see also the proof of Theorem 3.2 in [13]). It is easily seen that f is well defined and that it is a surjective homomorphism of anneids.…”

“…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 Graded Uj-Rings

“…By Theorem 2.2, the e-component of R/J g (R) is R e /J(R e ). According to the proof of Theorem 3.27 in[13] (see also Theorem 3.2 in[11]), we have that every nonzero homogeneous element from R/J g (R) can be identified with exactly one…”

“…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.…”