The graded Jacobson radical of associative rings.
Let \(G\) be a group or semigroup. A left module \(T\) over a \(G\)-graded ring \(R\) is a \(G\)-graded left module if there exist additive subgroups \(T_g\) of \(T\) such that \(T=\bigoplus_{g\in G}T_g\) and \(R_xT_y\subseteq T_{xy}\) for all \(x,y\in G\). A \(G\)-graded left module \(T\) over a \(G\)-graded ring \(R\) is a graded-simple module if \(T\
Gardner, B., Plant, A.
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Algebraic Properties of Semi-Direct Sums of Rings
Let R be an associative ring not necessarily with unity. We say that R is a semi-direct sum of rings S and I, if R = S + I, where S is a subring of a ring R, I is an ideal of R and S ∩ I = {0}.
Nowakowska Marta
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Jacobson radicals of Ore extensions
The proofs of Theorem 2 and Theorem 3 were ...
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The Jacobson Radical of a Semiring [PDF]
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