Projective and Injective Modules, An Introduction

Introduction

Modules, or unitary modules, can act as objects in a category, with R module homomorphisms acting as morphisms. Hence a projective module is merely a projective object within its category. Now if you're not familiar with category theory, don't panic; I'm going to describe projective and injective below. I just wanted you to know that these adjectives have general definitions that go beyond modules.

A module is projective if each homomorphism downstairs has a lift upstairs. Here is the technical definition.

A module P is projective if, for any pair of modules A and B, and any epimorphism f from A onto B, and any homomorphism g from P into B, there is at least one homomorphism h from P into A such that hf = g. A map from P into B lifts up to a compatible map from P into A.

Every free module is projective. See projective objects for a proof. As a special case, the ring R is always a projective R module.

A module J is injective if, for any pair of modules A and B, and any monomorphism f from A to B, and any homomorphism g from A to J, there is at least one homomorphism h from B to J such that fh = g.

Direct Sum/Product

Let P be the direct product or direct sum of modules C_i, where i ranges over an indexing set, and assume P is projective. We are given f(A) onto B, and g(C_i) into B. Extend g to all of P by mapping everything outside of C_i to 0. If a component other than i is nonzero, the image is 0. This is a valid homomorphism from P into B, and it has a lift h upstairs. Restrict h to C_i. The diagram commutes, and h(C_i) is the lift of g(C_i). Therefore each component C_i is projective.

If P is a direct sum we can prove the converse. Assume each C_i is projective. Given g(P) into B, realize that g defines, and is defined by, its action on each C_i. Lift each g(C_i) up to an h(C_i), and put these component functions together to build a composite function h. Verify that h is indeed a lift for g, hence P is projective.

Similar reasoning applies when P is injective. Given a function g(A) into C_i, extend it to all of P by setting the other components to 0. This implies a function h, which we restrict to C_i, and C_i is injective.

Conversely, if P is a direct sum or a direct product, and each C_i is injective, the function g(A) into P defines, and is defined by, component functions from A into each C_i. Each of these component functions implies a compatible function downstairs. Put these together to build a function h, and P is injective.

Unitary Modules

Unless otherwise stated, modules on this website are assumed unitary. But if M is not unitary, it is the cross product of a unitary submodule U and another submodule V that is killed by 1. Using the above, M is projective iff U and V are projective, and M is injective iff U and V are injective.