turning an ideal into a ring: division

Let I = < i_n > be an ideal which is contained in a proper ideal J = < j_n > of some (commutative) polynomial ring R.

J is a ring without a unit. I is an ideal of the ring J as well. Just as one can meaningfully divide polynomials in R with respect to one of its ideals (using Groebner Bases), can one divide polynomials in the ring J with respect to its ideal I?

In other words, is there, given a polynomial p in J, a notion of q = p (mod I) within the ring J? By this I mean

p = Sum(k) alpha_k * i_k + q

where all the alpha_k are polynomials in J.

If yes, can Singular compute the alpha's and q?

I do not quite understand the question. What do you impose on q?

In any case, the expression Sum(k) alpha_k * i_k is in I*J and you could
apply the Singular command division(p,I*J);

The result can then be used to get an expression as you propose,
with q the usual remainder of the division by I*J. Is this what you mean?