Free Modules, Projective, and Injective Modules

Abstract

Free modules are basic because they have bases. A right free module F over a ring R comes with a basis {e_{i} : i ∈ I} (for some indexing set I) so that every element in F can be uniquely written in the form Σ _iεI e _i r _i , where all but a finite number of the elements r_i ε R are zero. Free modules can also be described by a universal property, but the definition given above is more convenient for working inside the free module in question. We can also work with F by identifying it with R^(I), the direct sum of I copies of R (or more precisely R _R ). The direct sum R^(I) is contained (as a submodule) in the direct product R^I, which is usually “much bigger”: we have the equality R^(I) = R^I iff I is finite or R is the zero ring.