On two sequences of sets of mappings of abstract sets into a Dedekind ring
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ON TWO SEQUENCES OF SETS OF MAPPINGS OF ABSTRACT SETS INTO A DEDEKIND RING UDC 512.552.37+519.115
V. V. Skobelev
Abstract. This paper establishes some interrelations between a finite sequence of sets of mappings of an abstract set S into complete residue systems of pairwise relatively prime elements of any Dedekind ring and the corresponding sequence of sets of mappings of the set S into the complete residue system corresponding to the product of these elements. A relationship is revealed between the established interrelation and Lang’s theorem on isomorphic factor rings. A string model is presented that is an interpretation of structures investigated in the cases of the ring of integers and a one-element set S. It is shown that the results obtained can be used for computing the number of combinatorial objects defined in terms of finite residue rings. Keywords: Dedekind ring, mapping into complete residue systems, reversible matrix over a finite ring, Chinese residue theorem. INTRODUCTION A stable tendency of using algebraic models and methods in solving cryptography problems [1] ³s recently observed, for example, the use of computations performed in finite rings in constructing modern data encryption standards. Therefore, the development of combinatorial schemes [2] destined for the calculation of the number of objects constructed with the help of ring theory is topical. It is obvious that such a scheme can be represented in terms of mappings of abstract sets into the corresponding ring. This representation allows one to establish internal interrelations between ring theory [3–5], combinatorial analysis [2], and applied problems of information transformation, in particular, in cryptography. As a ring, a ring of the most general form can naturally be chosen that cover basic number-theoretic constructions used in modern cryptography. Among rings of this type is a Dedekind ring [3]. The objective of the present work is the investigation of interrelations between two sequences of sets of mappings of an abstract set into a Dedekind ring that are defined in terms of a complete system of residues to pairwise relatively prime moduli. In Sec. 1, the basic concepts are introduced and sequences of mappings being investigated are constructed. In Sec. 2, the equality of the cardinalities of structures constructed with the help of sequences of sets of mappings is proved. A link between these structures and Lang’s isomorphism theorem for factor rings [5] and the Chinese remainder theorem for Dedekind rings is established. The possibility of using such structures for calculating the number of invertible matrices over a finite numerical ring is shown. In Sec. 3, a string model as an interpretation of the structures being investigated is constructed for the case when an abstract set consists of one element and a Dedekind ring is the ring of integers. It is shown that, in terms of this model, one can investigate number-theoretic constructions based on the use of the Euler function and the Chinese remainder theorem for the ring of
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