Ring extension problem, Shukla cohomology and Ann-category theory

Abstract

Every ring extension of $A$ by $R$ induces a pair of group homomorphisms ${\mathcal{L}}^{\ast }:R\to En{d}_{\mathbb{Z}}(A)/L(A);{\mathcal{R}}^{\ast }:R\to En{d}_{\mathbb{Z}}(A)/R(A),$ preserving multiplication, satisfying some certain conditions. A such 4-tuple $(R,A,{\mathcal{L}}^{\ast },{\mathcal{R}}^{\ast })$ 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 $\mathbb{Z}$-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.

12 pages

References in corpus (1)

• Introduction to Ann-categories