Open AccessOn graded Ω-groups
Author(s) -
Emil Ilić-Georgijević
Publication year - 2015
Publication title -
filomat
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 34
eISSN - 2406-0933
pISSN - 0354-5180
DOI - 10.2298/fil1510167i
Subject(s) - mathematics , subring , graded ring , homogeneous , pure mathematics , isomorphism (crystallography) , ring theory , jacobson radical , discrete mathematics , combinatorics , ring (chemistry) , algebra over a field , commutative ring , chemistry , crystal structure , organic chemistry , crystallography , commutative property
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.