Abstract
In this chapter, we define formal matrix rings of order 2 and formal matrix rings of arbitrary order n. Their main properties are considered and examples of such rings are given. We indicate the relationship between formal matrix rings, endomorphism rings of modules, and systems of orthogonal idempotents of rings. For formal matrix rings, the Jacobson radical and the prime radical are described. We find when a formal matrix ring is Artinian, Noetherian, regular, unit-regular, and of stable rank 1. In the last section, clean and k-good matrix rings are considered.
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