ARMENDARIZ RINGS AND SEMICOMMUTATIVE RINGS

Abstract

In this note we concern the structures of Armendariz rings and semicommutative rings which are generalizations of reduced rings, the classical right quotient rings of Armendariz rings, the polynomial rings over semicommutative rings, and the relationships between Armendariz rings and semicommutative rings. We actually show that (i) for a right Ore ring R with Q its classical right quotient ring, R is Armendariz if and only if Q is Armendariz; (ii) for a semiprime right Goldie ring R with Q its classical right quotient ring, R is Armendariz , R is reduced , R is semicommutative , Q is Armendariz , Q is reduced , Q is semicommutative , Q is a finite direct product of division rings; (iii) there is a semicommutative ring over which the polynomial ring need not be semicommutative; and (iv) Armendariz rings need not be semicommutative. Moreover we extend the classes of Armendariz rings and semicommutative rings, observing the conditions under which some kinds of rings may be Armendariz or semicommutative.