"Factor-SAGBI Bases: a Tool for Computations in Subalgebras of Factor Algebras" Patrik Nordbeck. Department of Mathematics, Lund Box 118 Lund, S-221 00 Sweden E-mail:nordbeck@maths.lth.se We introduce canonical bases for subalgebras of quotients of polynomial rings. Canonical bases for subalgebras of polynomial rings were introduced by Kapur and Madlener, and independently by Robbiano and Sweedler. Using the language of Robbiano and Sweedler and referring to the ``non-quotient'' case as SAGBI bases theory (Subalgebra Analog to Groebner Bases for Ideals), we consequently call the canonical bases in our factor algebra setting Factor-SAGBI bases. SAGBI bases theory is (as the previous parenthesis indicates) strongly influenced by the theory of Groebner bases, introduced by Bruno Buchberger in his thesis; in e.g. the paper by Robbiano and Sweedler we find the notion of (subalgebra) reduction, the characterization (test) theorem using critical pairs (generalized S-polynomials), and the completion procedure of constructing bases. To make the theory work in our factor algebra setting, we need just complete the SAGBI theory at a few points. We try, as far as possible, to work in the normal complements of the ideals we factor out, so e.g. our subalgebra reduction also includes the usual Groebner basis reduction. In the test and construction of our bases we are forced to consider, besides critical pairs, one additional type of element.