Basic Concepts on Rings and Modules

Abstract

All rings we will consider will be associative rings with identity, and all modules will be unital modules. They will be right modules unless otherwise specified. Since we want all modules MR to have an endomorphism ring End(MR) (the zero module as well), the identity of the ring may be equal to zero, in which case the ring trivially reduces to zero. Ring homomorphisms send the identity of the domain to the identity of the codomain, and subrings must have the same identity of the ring. Whenever we will say “ideal,” we will mean a “two-sided ideal,” though we will sometimes say “a two-sided ideal” in order to emphasize it.