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.