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تسجيل مستخدم جديدCrossed Bimodules over Rings and Shukla Cohomology
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In this paper we present some applications of Ann-category theory to classification of crossed bimodules over rings, classification of ring extensions of the type of a crossed bimodule.
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Every ring extension of $A$ by $R$ induces a pair of group homomorphisms $mathcal{L}^{*}:Rto End_Z(A)/L(A);mathcal{R}^{*}:Rto End_Z(A)/R(A),$ 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.
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