Alpha - Skew Pi - Armendariz Rings
Abstract
In this article we introduce a new concept called Alpha-skew Pi-Armendariz rings (Alpha - S Pi - AR)as a generalization of the notion of Alpha-skew Armendariz rings.Another important goal behind studying this class of rings is to employ it in order to design a modern algorithm of an identification scheme according to the evolution of using modern algebra in the applications of the field of cryptography.We investigate general properties of this concept and give examples for illustration. Furthermore, this paperstudy the relationship between this concept and some previous notions related to Alpha-skew Armendariz rings. It clearly presents that every weak Alpha-skew Armendariz ring is Alpha-skew Pi-Armendariz (Alpha-S Pi-AR). Also, thisarticle showsthat the concepts of Alpha-skew Armendariz rings and Alpha-skew Pi- Armendariz rings are equivalent in case R is 2-primal and semiprime ring.Moreover, this paper proves for a semicommutative Alpha-compatible ringR that if R[x;Alpha] is nil-Armendariz, thenR is an Alpha-S Pi-AR. In addition, if R is an Alpha - S Pi -AR, 2-primal and semiprime ring, then N(R[x;Alpha])=N(R)[x;Alpha]. Finally, we look forwardthat Alpha-skew Pi-Armendariz rings (Alpha-S Pi-AR)be more effect (due to their properties) in the field of cryptography than Pi-Armendariz rings, weak Armendariz rings and others.For these properties and characterizations of the introduced concept Alpha-S Pi-AR, we aspire to design a novel algorithm of an identification scheme.
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