I'm teaching a variety of undergraduate and graduate geometry classes (mostly for in-service teachers) which range from elementary axiomatic geometry to more advanced transformational geometry. I'm looking to bring various levels of abstract algebra into these classes that incorporate basic ring and field theory to bridge the gap between these two traditional approaches to geometry but without teaching a full algebraic geometry course. In particular, I'm looking for suggestions of textbooks that give a good treatment of:
- Abstract affine spaces including finite and rational affine spaces,
- The relationship between affine spaces defined using abstract algebra and affine spaces defined by axioms (particularly affine planes and 3-dimensional affine spaces),
- Projective spaces using homogeneous coordinates from affine spaces.
Are there any books out there that cover this material in a way that doesn't get too heavy into advanced algebraic geometry?