ABSTRACT
In this chapter we present some of the basics of the classical theory of noncommutative rings. In my mind no one can improve on the elegant exposition in Herstein’s monograph Noncommutative Rings [H]. So this chapter follows closely the exposition of parts of this monograph. Throughout, the symbol R M will denote the fact that R is a (not necessarily commutative) ring and that M is a left R-module. All R-modules will be left R-modules, but we will show that the theory we develop is right-left symmetric. Recall that (0 : M), the annihilator of M, is the ideal consisting of all r ∈ R such that rM = < 0 > . For m ∈ M, also recall that (0 : m), the annihilator of m, is the left ideal consisting of all r ∈ R such that rm = 0.
