Chapter 8: invariant preserving linear matrix mappings over rings and related topics

This paper surveys the ideas involved in the theory of invariant preserving linear mappings of matrixrings where the scalar ring is not necessarily a field. Section 1 provides several historical examples of the origins of these problems. Section 2 discusses the basic context when the vector space over a field is replaced by a projective module over a commutative ring. Section 3 sketches the classification of the rank one preserving linear mappings using the approach of McDonald, Marcus, and Moyls. Section 4 continues the discussion of Section 3 by placing the problem within the context of group schemes and the invariant preserving theory of Waierhousc. Section 5 begins a sketch of the evolution of these ideas to a context where the scalar ring h not necessarily commutative with a discussion of some classical results of Hua concerning coherence, projective geometry, and matrices over division rings. Trie concluding section, Section 6, discusses the results developed by Wong of linear preserving maps over noncommutative scaiar rings.