It is proved that radical of every generalized ring is a G-graded ideal when its index is an abelian group G. The relation between radicals of generalized rings and Γ-rings is given. The explicit formulas for generalized ring A and radical properties r = rb, rl, rj , rn are obtained: r(A) = g.m.r(A) = ∑ {r(Aij) | i, j ∈ I}. The relation between the radicals of path algebras and connectivity of ...

The aim of this study is to characterize rings having the following properties for a non-trivial idempotent element e R, U (eRe) = + eJ(R)e J (and N (eRe)),where (-), (-) and denote group units, set all nilpotent elements Jacobson radical respectively. In present paper, some characterizations are also obtained in terms every form u, where e2 ∈ R u U(eRe).

We extend the notion of the Leavitt path algebra of a graph E to include all directed graphs. We show how various ring-theoretic properties of these more general structures relate to the corresponding properties of Leavitt path algebras of row-finite graphs. Specifically, we identify those graphs for which the corresponding Leavitt path algebra is simple; purely infinite simple; exchange; and s...