Module (mathematics) - Types of Modules

Types of Modules

Finitely generated. An R-module M is finitely generated if there exist finitely many elements x₁,...,x_n in M such that every element of M is a linear combination of those elements with coefficients from the ring R.

Cyclic. A module is called a cyclic module if it is generated by one element.

Free. A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces.

Projective. Projective modules are direct summands of free modules and share many of their desirable properties.

Injective. Injective modules are defined dually to projective modules.

Flat. A module is called flat if taking the tensor product of it with any short exact sequence of R-modules preserves exactness.

Simple. A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.

Semisimple. A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible.

Indecomposable. An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable, but there are indecomposable modules which are not simple (e.g. uniform modules).

Faithful. A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. rx ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.

Noetherian. A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.

Artinian. An Artinian module is a module which satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.

Graded. A graded module is a module with a decomposition as a direct sum M = ⊕_x M_x over a graded ring R = ⊕_x R_x such that R_xM_y ⊂ M_{x + y} for all x and y.

Uniform. A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.