On the division algorithm with remain in a ring of polynomials of N-variables

As is known, the ring of polynomials in one variable is a Euclidean ring, which means that for any two polynomials there is also a unique pair of polynomials for which or r(x)=0. In this article, we will consider a generalization of the procedure for finding the quotient and remainder to the case of a ring . As we will see, in this case, the situation is not as clear-cut as in