On graded $\Omega$-groups

Abstract

In this paper we study the notion of a graded
$\Omega$-group $(X,+,\Omega),$ 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 $\Omega$ and the grading of
$(X,+)$ induce operations (generally partial) on the grading set.
We prove that graded $\Omega$-groups in Krasner's sense are
determined up to isomorphism by their homogeneous parts, which,
with respect to induced operations, represent partial structures
called $\Omega$-\emph{homogroupoids}, thus narrowing down the
theory of graded $\Omega$-groups to the theory of
$\Omega$-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
$\Omega$-homogroupoids thus extending results of A.~V.~Mikhalev,
I.~N.~Balaba and S.~A.~Pikhtilkov in a natural way. We study some
classes of $\Omega$-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.