Project 1: Give an example of a principal ideal domain which is not a Euclidean domain.
Project 2: Tsen-Lang theory (C_n fields).
Project 3: Wedderburn's Little Theorem (about associative division rings of finite order) and Artin-Zorn Theorem (generalization to alternative division rings).
Project 4: Effective Nullstellensatz. Possible starting point: MR0944576 (89h:12008).
Project 5: The fibre dimension theorem. (One possible source for this is Basic Algebraic Geometry by Shafarevich.)
Project 6: Hilbert's theorem on the finite generation of the ring of invariants.
Project 7: Krull dimension.
Project 8: Efficient implementations of Buchsberger's algorithm Possible sources: Becker and Weispfenning MR1213453, Cox, Little and O'Shea MR1189133.
Project 9: Application of Grobner bases. Start with section 2.8 of Cox, Little and O'Shea (see the reference above).
Project 10: SAGBI = Sabalgebra analogue of Grobner bases for ideals. Possible sources: my paper MR1966757 and the references there.