The remainder theorem and factor theorem for polynomials over noncommutative coefficient rings

Abstract

Let R be an associative ring which has a multiplicative identity element but need not be commutative. Let and α ∈ R. It is known that there exist uniquely determined and and r,s ∈ R such that . Examples show that q(X) and p(X) need not coincide. As in the classical Remainder Theorem over a commutative ring, , but . As consequences, a pair of generalizations of the classical Factor Theorem is obtained, as well as a new characterization of commutative rings. This note could find classroom/homework use in a course on abstract algebra as enrichment material for the unit on rings and polynomials.