### Abstract:

The applications of generalized inverses of matrices appear in many ﬁelds like applied mathematics, statistics and engineering [2]. In this thesis, we discuss generalized inverses of matrices over Ore polynomial rings (also called Ore matrices).
We ﬁrst introduce some necessary and suﬃcient conditions for the existence of {1}-, {1,2}-, {1,3}-, {1,4}- and MP-inverses of Ore matrices, and give some explicit formulas for these inverses. Using {1}-inverses of Ore matrices, we present the solutions of linear systems over Ore polynomial rings. Next, we extend Roth's Theorem 1 and generalized Roth's Theorem 1 to the Ore matrices case. Furthermore, we consider the extensions of all the involutions ψ on R(x), and construct some necessary and suﬃcient conditions for ψ to be an involution on R(x)[D;σ,δ]. Finally, we obtain two diﬀerent explicit formulas for {1,3}- and {1,4}-inverses of Ore matrices.
The Maple implementations of our main algorithms are presented in the Appendix.