Homogeneously simple associative algebras.
Suppose that \(G\) is group and \(A=\bigoplus_{g\in G}A_g\) is a \(G\)-graded algebra over a field \(k\). It is assumed that in \(A\) for any elements \(a_i\in A_{g_i}\), \(i=1,2,3,\) there exists a non-zero element \(\lambda(g_1,g_2,g_3)\in k\) such that \((a_1a_2)a_3=\lambda(g_1,g_2,g_3)a_1(a_2a_3)\). Suppose that the field \(k\) is algebraically closed and \(A\) has no nontrivial homogeneous ideals. Then \(A=B\text{Mat}(n,k)\), where \(B\) is a homogeneous field and \(\text{Mat}(n,k)\) is a full matrix algebra of size \(n\) over \(k\).
scientific article; zbMATH DE number 5975538