### 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.

#### Full Text:

PDF### Refbacks

- There are currently no refbacks.