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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 proving the independence of the axiomatics of Ann-category definition. And then we have proved that after adding an axiom into the definition of categorical rings, we obtain the new axiomatics which is equivalent to the one of Ann-categories.
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
This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$A=(A,{Ah},c,(0,g,d),a,(1,l,r),{Lh},{Rh})$$ be an Ann-category. The coherence theorem states that in t
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 A
In this paper, we show another proof of the problem by constructing a strict monoidal category M(C) consisting of M-functors and M-morphisms of a category C and we prove C is equivalent to it. The proof is based on a basic character of monoidal equiv