| Abstract | In this paper we generalize some basic applications of Groebner bases in commutative polynomial rings to the non-commutative case. We define a non-commutative elimination order. Methods of finding the intersection of two ideals are given. If both the ideals are monomial we deduce a finitely written basis for their intersection. We find the kernel of a homomorphism, and decide membership of the image. Finally we show how to obtain a Groebner basis for an ideal by considering a related homogeneous ideal.
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