On graded Ω-groups

Abstract

In this paper we study the notion of a graded ?-group (X;+ ?), but graded in the sense of M. Krasner, i.e., we impose nothing on the grading set except that it is nonempty, since operations of and the grading of (X,+) induce operations (generally partial) on the grading set. We prove that graded ?-groups in Krasner?s sense are determined up to isomorphism by their homogeneous parts, which, with respect to induced operations, represent partial structures called ?-homogroupoids, thus narrowing down the theory of graded -groups to the theory of ?-homogroupoids. This approach already proved to be useful in questions regarding A. V. Kelarev?s S-graded rings inducing S; where S is a partial cancellative groupoid. Particularly, in this paper we prove that the homogeneous subring of a Jacobson S-graded ring inducing S is Jacobson under certain assumptions. We also discuss the theory of prime radicals for ?-homogroupoids thus extending results of A. V. Mikhalev, I. N. Balaba and S. A. Pikhtilkov in a natural way. We study some classes of ?-homogroupoids for which the lower and upper weakly solvable radicals coincide and also, study the question of the homogeneity of the prime radical of a graded ring.

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

“…Let e ∈ E(S) * . By Theorem 8.17 in [8], we have that P g (R) ∩ R e ⊆ P (R e ), where P (R e ) denotes the classical prime radical of R e . However, J g (R) ∩ R e = J(R e ) by Theorem 2.2.…”

“…Let us recall from [8] that an S-graded ring R inducing S is said to be graded prime if for every two homogeneous ideals A, B of R, the condition AB = 0 implies ON GRADED U J-RINGS…”

“…A homogeneous ideal A of R is said to be graded prime if R/A is graded prime. 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].…”

“…We may use this characterization in order to define a U U -ring which does not necessarily have an identity, but also to define the notion of a graded U U -ring. Let us recall from [8] that an S-graded ring R inducing S is said to be graded prime if for every two homogeneous ideals A, B of R, the condition AB = 0 implies ON GRADED U J-RINGS…”