Ring extension problem, Shukla cohomology and Ann-category theory

TL;DR

This paper explores how ring extensions relate to cohomology and Ann-category theory, introducing a new application of Ann-category in algebraic extensions and cohomology via Shukla cohomology.

Contribution

It establishes a connection between ring extensions, Shukla cohomology, and Ann-category structures, providing a novel application in algebraic extension problems.

Findings

Ring extensions induce specific group homomorphisms and bimodule structures.

Obstructions in cohomology classify Ann-category structures.

First application of Ann-category in algebraic extension and cohomology theories.

Abstract

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.