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In this paper we study the structure of a class of categories having two operations which satisfy axioms analoguos to that of rings. Such categories are called Ann - categories. We obtain the classification theorems for regular Ann - categories and Ann - functors by using Mac Lane - Shukla cohomology of rings. These results give new interpretations of the cohomology groups and of the rings
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
Each Ann-category $A$ is equivalent to an Ann-category of the type $(R,M),$ where $M$ is an $R$-bimodule. The family of constraints of $A$ induces a {it structure} on $(R,M).$ The main result of the paper is: 1. {it There exists a bijection between
This paper presents the structure conversion by which from an Ann-category $A,$ we can obtain its reduced Ann-category of the type $(R,M)$ whose structure is a family of five functions $k=(xi,eta,alpha,lambda,rho)$. Then we will show that each Ann-ca
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.
In this paper, we have studied the axiomatics of {it Ann-categories} and {it categorical rings.} These are the categories with distributivity constraints whose axiomatics are similar with those of ring structures. The main result we have achieved is