# Glossary of module theory

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Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.

## Basic definition

- left R-module
- A left module over the ring is an abelian group with an operation (called scalar multipliction) satisfies the following condition:

- right R-module
- A right module over the ring is an abelian group with an operation satisfies the following condition:

- Or it can be defined as the left module over (the opposite ring of ).

- bimodule
- If an abelian group is both a left -module and right -module, it can be made to a -bimodule if .

- submodule
- Given is a left -module, a subgroup of is a submodule if .

- homomorphism of -modules
- For two left -modules , a group homomorphism is called homomorphism of -modules if .

- quotient module
- Given a left -modules , a submodule , can be made to a left -module by . It is also called a
**factor module**.

- annihilator
- The annihilator of a left -module is the set . It is a (left) ideal of .
- The annihilator of an element is the set .

## Types of modules

- finitely generated module
- A module is finitely generated if there exist finitely many elements in such that every element of is a finite linear combination of those elements with coefficients from the scalar ring .

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

- free module
- A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring .

- Projective module
- A -module is called a projective module if given a -module homomorphism , and a surjective -module homomorphism , there exists a -module homomorphism such that .

- The following conditions are equivalent:

- The covariant functor is exact.
- is a projective module.
- Every short exact sequence is split.
- is a direct summand of free modules.

- In particular, every free module is projective.

- injective module
- A -module is called an injective module if given a -module homomorphism , and an injective -module homomorphism , there exists a

-module homomorphism such that .

- The following conditions are equivalent:

- The contravariant functor is exact.
- is a injective module.
- Every short exact sequence is split.

- flat module
- A -module is called a flat module if the tensor product functor is exact.
- In particular, every projective module is flat.

- simple module
- A simple module is a nonzero module whose only submodules are zero and itself.

- indecomposable module
- 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.

- principal indecomposable module
- A cyclic indecomposable projective module is known as a PIM.

- semisimple module
- A module is called semisimple if it is the direct sum of simple submodules.

- faithful module
- A faithful module is one where the action of each nonzero on is nontrivial (i.e. for some x in M). Equivalently, is the zero ideal.

- Noetherian module
- A Noetherian module is a module such that every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps.

- Artinian module
- An Artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.

- finite length module
- A module which is both Artinian and Noetherian has additional special properties.

- graded module
- A module over a graded ring is a graded module if can be expressed as a direct sum and .

- invertible module
- Roughly synonymous to rank 1 projective module.

- uniform module
- Module in which every two non-zero submodules have a non-zero intersection.

- algebraically compact module (pure injective module)
- Modules in which all systems of equations can be decided by finitary means. Alternatively, those modules which leave pure-exact sequence exact after applying Hom.

- injective cogenerator
- An injective module such that every module has a nonzero homomorphism into it.

- irreducible module
- synonymous to "simple module"

- completely reducible module
- synonymous to "semisimple module"

## Operations on modules

- Essential extension
- An extension in which every nonzero submodule of the larger module meets the smaller module in a nonzero submodule.

- Injective envelope
- A maximal essential extension, or a minimal embedding in an injective module

- Projective cover
- A minimal surjection from a projective module.

- Socle
- The largest semisimple submodule

- Radical of a module
- The intersection of the maximal submodules. For Artinian modules, the smallest submodule with semisimple quotient.

### Changing scalars

- Restriction of scalars
- Uses a ring homomorphism from
*R*to*S*to convert*S*-modules to*R*-modules

- Extension of scalars
- Uses a ring homomorphism from
*R*to*S*to convert*R*-modules to*S*-modules

- Localization of a module
- Converts
*R*modules to*S*modules, where*S*is a localization of*R*

- Endomorphism ring
- A left
*R*-module is a right*S*-module where*S*is its endomorphism ring.

## Homological algebra

## Modules over special rings

- D-module
- A module over a ring of differential operators.
- Drinfeld module
- A module over a ring of functions on algebraic curve with coefficients from a finite field.
- Galois module
- A module over the group ring of a Galois group
- Structure theorem for finitely generated modules over a principal ideal domain
- Finitely generated modules over PIDs are finite direct sums of primary cyclic modules.
- Tate module
- A special kind of Galois module

## Miscellaneous

- Rational canonical form
- elementary divisor
- invariants
- fitting ideal
- normal forms for matrices
- Jordan Hölder composition series
- tensor product

## See also

## References

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