ASSOCIATIVITY OF (for fi, f) POLYNOMIAL MODULE

Abstract

Unless stated otherwise R will stand for a unitary ring, not necessarily associative, and M for a right unitary R-module, for which the associative property (i.c., a (ab) = (aa)b for all & in M and a,b in R) is not assumed. If R is associative, the ordinary polynomial ring R [x] is associative and in obvious notation M [x] is also associative as right R[x]-module, whenever M is associative. In [2] the construction of an (fo. f. ft) extension ring S = R [x: f f fi of R was given, and its associativity was discussed. Even if R is associative, it is not necessary that S is associative. However, certain necessary and sufficient conditions were found in [2] in order that S be associative. In [1] the construction of a polynomial module NM [x: fo. fi of M was taken up and it was shown that N is a right S-module. If R and M are associative, then the problem of associativity of N arises. In this paper we intend to lay down conditions under which N is ensured to be associative. Indeed, these conditions look like those as in [1] and they are also studied in [3].