Classification of pairs of mutually annihilating operators in a graded space and representations of the diad of generalized uniserial algebras

Let $A_1$ and $A_2$ be semiperfect rings with Jacobson radicals $R_1$ and $R_2$ where by $A_1/R_1\cong A_2/R_2\cong T$ and suppose there are given isomorphisms $\varphi_1\colon A_i\to T$ ($i=1,2$). The diad of the rings $A_1$ and $A_2$ with common factor ring $T$ is the ring $A_1\times_TA_2$ consisting of all $(a_1,a_2)\in A_1\times A_2$ for which $\varphi(a_1)=\varphi(a_2)$. Representations of the dyad of generalized uniserial algebras over an algebraically closed field are described in the paper. Bibl. 9 titles.