Ring extension problem, Shukla cohomology and Ann-category theory

Every ring extension of $A$ by $R$ induces a pair of group homomorphisms $\mathcal{L}^{}:R\to End_\Z(A)/L(A);\mathcal{R}^{}:R\to End_\Z(A)/R(A),$ preserving multiplication, satisfying some certain conditions. A such 4-tuple $(R,A,\mathcal{L}^{},\mathcal{R}^{})$ is called a ring pre-extension. Each ring pre-extension induces a $R$-bimodule structure on bicenter $K_A$ of ring $A,$ and induces an obstruction $k,$ which is a 3-cocycle of $\Z$-algebra $R,$ with coefficients in $R$-bimodule $K_A$ in the sense of Shukla. Each obstruction $k$ in this sense induces a structure of a regular Ann-category of type $(R,K_A).$ This result gives us the first application of Ann-category in extension problems of algebraic structures, as well as in cohomology theories.

💡 Research Summary

The paper addresses the classical problem of extending a ring A by another ring R, but does so from a novel perspective that intertwines Shukla cohomology with the theory of Ann‑categories. The authors begin by observing that any ring extension yields two canonical group homomorphisms

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