Free Modules, Projective, and Injective Modules

Abstract

AbstractFree modules are basic because they have bases. A right free module F over a ring R comes with a basis {ei} : 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 ri 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 RI, which is usually “much bigger”: we have the equality R(I) = RI iff I is finite or R is the zero ring.KeywordsPrime IdealDirect SummandCommutative RingProjective ModuleFree ModuleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.