Let \(k\) be a field (of any characteristic), and let \(R\) be a \(k\)-algebra. Suppose that \(R\) is equipped with a grading by a (multiplicative, finite or infinite) group \(G\), that is, a decomposition \(R = \bigoplus_{g \in G} R_g\) of \(R\) into \(k\)-subvectorspaces \(R_g\) which is compatible with the multiplication operation in the sense that \(R_g R_h \subseteq R_{gh}\) for all \(g,h \in G\). The graded Jacobson radical \(J^{gr}(R)\) of \(R\) is, by definition, the intersection of the annihilators of all graded irreducible left \(R\)-modules.
If the graded Jacobson radical vanishes and \(\dim_k (R) < \infty\) then \(R\) is a direct sum of finitely many graded simple algebras; this is the content of Theorem 2.4 of the paper under review. The graded simple algebras which are graded left Noetherian are completely described in Theorem 2.5.
Of course one can also forget about the grading and consider the (usual, ungraded) Jacobson radical. If the grading has finite support (so that \(R_g = \{0\}\) for all but finitely many \(g\)), it is known that the graded Jacobson radical is contained in the (usual, ungraded) Jacobson radical of \(R\), but in general the connection between the two notions far from clear.
The main result in the paper under review is Theorem 3.4, stating that if \(R\) is a \(G\)-graded algebra of block-triangular matrices with entries in \(K\) then the (usual, ungraded) Jacobson radical coincides with the graded Jacobson radical \(J^{gr}(R)\). This leads to a complete classification of all possible \(G\)-gradings of \(R\) in Corollary 3.6; this classification is given in terms of division gradings on (full) matrix algebras, finite collections of elements of \(G\), and a standard construction of gradings on block-triangular matrix algebras as detailed at the end of §2 of the paper.