Glossary of module theory

Short description: none

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

See also: Glossary of linear algebra, Glossary of ring theory, Glossary of representation theory.

A

algebraically compact

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

annihilator

1.  The annihilator of a left [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ M }[/math] is the set [math]\displaystyle{ \textrm{Ann}(M) := \{ r \in R ~|~ rm = 0 \, \forall m \in M \} }[/math] . It is a (left) ideal of [math]\displaystyle{ R }[/math].

2.  The annihilator of an element [math]\displaystyle{ m \in M }[/math] is the set [math]\displaystyle{ \textrm{Ann}(m) := \{ r \in R ~|~ rm = 0 \} }[/math].

Artinian

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

associated prime

1.  associated prime

automorphism

An automorphism is an endomorphism that is also an isomorphism.

Azumaya

Azumaya's theorem says that two decompositions into modules with local endomorphism rings are equivalent.

B

balanced

balanced module

basis

A basis of a module [math]\displaystyle{ M }[/math] is a set of elements in [math]\displaystyle{ M }[/math] such that every element in the module can be expressed as a finite sum of elements in the basis in a unique way.

Beauville–Laszlo

Beauville–Laszlo theorem

big

"big" usually means "not-necessarily finitely generated".

bimodule

bimodule

C

canonical module

canonical module (the term "canonical" comes from canonical divisor)

category

The category of modules over a ring is the category where the objects are all the (say) left modules over the given ring and the morphisms module homomorphisms.

character

character module

chain complex

chain complex (frequently just complex)

closed submodule

A module is called a closed submodule if it does not contain any essential extension.

Cohen–Macaulay

Cohen–Macaulay module

coherent

A coherent module is a finitely generated module whose finitely generated submodules are finitely presented.

cokernel

The cokernel of a module homomorphism is the codomain quotiented by the image.

compact

A compact module

completely reducible

Synonymous to "semisimple module".

completion

completion of a module

composition

Jordan Hölder composition series

continuous

continuous module

countably generated

A countably generated module is a module that admits a generating set whose cardinality is at most countable.

cyclic

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

D

D

A D-module is a module over a ring of differential operators.

decomposition

A decomposition of a module is a way to express a module as a direct sum of submodules.

dense

dense submodule

determinant

The determinant of a finite free module over a commutative ring is the r-th exterior power of the module when r is the rank of the module.

differential

A differential graded module or dg-module is a graded module with a differential.

direct sum

A direct sum of modules is a module that is the direct sum of the underlying abelian group together with component-wise scalar multiplication.

dual module

The dual module of a module M over a commutative ring R is the module [math]\displaystyle{ \operatorname{Hom}_R(M, R) }[/math].

dualizing

dualizing module

Drinfeld

A Drinfeld module is a module over a ring of functions on algebraic curve with coefficients from a finite field.

E

Eilenberg–Mazur

Eilenberg–Mazur swindle

elementary

elementary divisor

endomorphism

1.  An endomorphism is a module homomorphism from a module to itself.

2.  The endomorphism ring is the set of all module homomorphisms with addition as addition of functions and multiplication composition of functions.

enough

enough injectives

enough projectives

essential

Given a module M, an essential submodule N of M is a submodule that every nonzero submodule of M intersects non-trivially.

exact

exact sequence

Ext functor

Ext functor

extension

Extension of scalars uses a ring homomorphism from R to S to convert R-modules to S-modules.

F

faithful

A faithful module [math]\displaystyle{ M }[/math] is one where the action of each nonzero [math]\displaystyle{ r \in R }[/math] on [math]\displaystyle{ M }[/math] is nontrivial (i.e. [math]\displaystyle{ rx \ne 0 }[/math] for some [math]\displaystyle{ x }[/math] in [math]\displaystyle{ M }[/math]). Equivalently, [math]\displaystyle{ \textrm{Ann}(M) }[/math] is the zero ideal.

finite

The term "finite module" is another name for a finitely generated module.

finite length

A module of finite length is a module that admits a (finite) composition series.

finite presentation

1.  A finite free presentation of a module M is an exact sequence [math]\displaystyle{ F_1 \to F_0 \to M }[/math] where [math]\displaystyle{ F_i }[/math] are finitely generated free modules.

2.  A finitely presented module is a module that admits a finite free presentation.

finitely generated

A module [math]\displaystyle{ M }[/math] is finitely generated if there exist finitely many elements [math]\displaystyle{ x_1,...,x_n }[/math] in [math]\displaystyle{ M }[/math] such that every element of [math]\displaystyle{ M }[/math] is a finite linear combination of those elements with coefficients from the scalar ring [math]\displaystyle{ R }[/math].

fitting

1.  fitting ideal

2.  Fitting's lemma

five

Five lemma

flat

A [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ F }[/math] is called a flat module if the tensor product functor [math]\displaystyle{ - \otimes_R F }[/math] is exact.

In particular, every projective module is flat.

free

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 [math]\displaystyle{ R }[/math].

Frobenius reciprocity

Frobenius reciprocity.

G

Galois

A Galois module is a module over the group ring of a Galois group.

generating set

A subset of a module is called a generating set of the module if the submodule generated by the set (i.e., the smallest subset containing the set) is the entire module itself.

global

global dimension

graded

A module [math]\displaystyle{ M }[/math] over a graded ring [math]\displaystyle{ A = \bigoplus_{n\in \mathbb N}A_n }[/math] is a graded module if [math]\displaystyle{ M }[/math] can be expressed as a direct sum [math]\displaystyle{ \bigoplus_{i\in \mathbb N}M_i }[/math] and [math]\displaystyle{ A_i M_j \subseteq M_{i+j} }[/math].

H

Herbrand quotient

A Herbrand quotient of a module homomorphism is another term for index.

Hilbert

1.  Hilbert's syzygy theorem

2.  The Hilbert–Poincaré series of a graded module.

3.  The Hilbert–Serre theorem tells when a Hilbert–Poincaré series is a rational function.

homological dimension

homological dimension

homomorphism

For two left [math]\displaystyle{ R }[/math]-modules [math]\displaystyle{ M_1, M_2 }[/math], a group homomorphism [math]\displaystyle{ \phi: M_1 \to M_2 }[/math] is called homomorphism of [math]\displaystyle{ R }[/math]-modules if [math]\displaystyle{ r \phi(m) = \phi (r m) \, \forall r \in R, m \in M_1 }[/math] .

Hom

Hom functor

I

idempotent

An idempotent is an endomorphism whose square is itself.

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 not conversely).

index

The index of an endomorphism [math]\displaystyle{ f : M \to M }[/math] is the difference [math]\displaystyle{ \operatorname{length}(\operatorname{coker}(f)) - \operatorname{length}(\operatorname{ker}(f)) }[/math], when the cokernel and kernel of [math]\displaystyle{ f }[/math] have finite length.

injective

1.  A [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ Q }[/math] is called an injective module if given a [math]\displaystyle{ R }[/math]-module homomorphism [math]\displaystyle{ g: X \to Q }[/math], and an injective [math]\displaystyle{ R }[/math]-module homomorphism [math]\displaystyle{ f: X \to Y }[/math], there exists a [math]\displaystyle{ R }[/math]-module homomorphism [math]\displaystyle{ h : Y \to Q }[/math] such that [math]\displaystyle{ f \circ h = g }[/math].

The module Q is injective if the diagram commutes

The following conditions are equivalent:

• The contravariant functor [math]\displaystyle{ \textrm{Hom}_R( - , I) }[/math] is exact.
• [math]\displaystyle{ I }[/math] is a injective module.
• Every short exact sequence [math]\displaystyle{ 0 \to I \to L \to L' \to 0 }[/math] is split.

2.  An injective envelope (also called injective hull) is a maximal essential extension, or a minimal embedding in an injective module.

3.  An injective cogenerator is an injective module such that every module has a nonzero homomorphism into it.

invariant

invariants

invertible

An invertible module over a commutative ring is a rank-one finite projective module.

irreducible module

Another name for a simple module.

isomorphism

An isomorphism between modules is an invertible module homomorphism.

J

Jacobson

density theorem

K

Kähler differentials

Kähler differentials

Kaplansky

Kaplansky's theorem on a projective module says that a projective module over a local ring is free.

kernel

The kernel of a module homomorphism is the pre-image of the zero element.

Koszul complex

Koszul complex

Krull–Schmidt

The Krull–Schmidt theorem says that (1) a finite-length module admits an indecomposable decomposition and (2) any two indecomposable decompositions of it are equivalent.

L

length

The length of a module is the common length of any composition series of the module; the length is infinite if there is no composition series. Over a field, the length is more commonly known as the dimension.

linear

1.  A linear map is another term for a module homomorphism.

2.  Linear topology

localization

Localization of a module converts R modules to S modules, where S is a localization of R.

M

Matlis module

Matlis module

Mitchell's embedding theorem

Mitchell's embedding theorem

Mittag-Leffler

Mittag-Leffler condition (ML)

module

1.  A left module [math]\displaystyle{ M }[/math] over the ring [math]\displaystyle{ R }[/math] is an abelian group [math]\displaystyle{ (M, +) }[/math] with an operation [math]\displaystyle{ R \times M \to M }[/math] (called scalar multipliction) satisfies the following condition:

[math]\displaystyle{ \forall r,s \in R, \forall m,n \in M }[/math],

1. [math]\displaystyle{ r (m + n) = rm + rn }[/math]
2. [math]\displaystyle{ r (s m) = (r s) m }[/math]
3. [math]\displaystyle{ 1_R \, m = m }[/math]

2.  A right module [math]\displaystyle{ M }[/math] over the ring [math]\displaystyle{ R }[/math] is an abelian group [math]\displaystyle{ (M, +) }[/math] with an operation [math]\displaystyle{ M \times R \to M }[/math] satisfies the following condition:

[math]\displaystyle{ \forall r,s \in R, \forall m,n \in M }[/math],

1. [math]\displaystyle{ (m + n) r = m r + n r }[/math]
2. [math]\displaystyle{ (m s) r = r (s m) }[/math]
3. [math]\displaystyle{ m 1_R = m }[/math]

3.  All the modules together with all the module homomorphisms between them form the category of modules.

module spectrum

A module spectrum is a spectrum with an action of a ring spectrum.

N

nilpotent

A nilpotent endomorphism is an endomorphism, some power of which is zero.

Noetherian

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.

normal

normal forms for matrices

P

perfect

1.  perfect complex

2.  perfect module

principal

A principal indecomposable module is a cyclic indecomposable projective module.

primary

primary submodule

projective

The characteristic property of projective modules is called lifting.

A [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ P }[/math] is called a projective module if given a [math]\displaystyle{ R }[/math]-module homomorphism [math]\displaystyle{ g: P \to M }[/math], and a surjective [math]\displaystyle{ R }[/math]-module homomorphism [math]\displaystyle{ f: N \to M }[/math], there exists a [math]\displaystyle{ R }[/math]-module homomorphism [math]\displaystyle{ h : P \to N }[/math] such that [math]\displaystyle{ f \circ h = g }[/math].

The following conditions are equivalent:

• The covariant functor [math]\displaystyle{ \textrm{Hom}_R(P, - ) }[/math] is exact.
• [math]\displaystyle{ M }[/math] is a projective module.
• Every short exact sequence [math]\displaystyle{ 0 \to L \to L' \to P \to 0 }[/math] is split.
• [math]\displaystyle{ M }[/math] is a direct summand of free modules.

In particular, every free module is projective.

2.  The projective dimension of a module is the minimal length of (if any) a finite projective resolution of the module; the dimension is infinite if there is no finite projective resolution.

3.  A projective cover is a minimal surjection from a projective module.

pure submodule

pure submodule

Q

Quillen–Suslin theorem

The Quillen–Suslin theorem states that a finite projective module over a polynomial ring is free.

quotient

Given a left [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ M }[/math] and a submodule [math]\displaystyle{ N }[/math], the quotient group [math]\displaystyle{ M/N }[/math] can be made to be a left [math]\displaystyle{ R }[/math]-module by [math]\displaystyle{ r(m+N) = rm + N }[/math] for [math]\displaystyle{ r \in R, \, m \in M }[/math]. It is called a quotient module or factor module.

R

radical

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

rational

rational canonical form

reflexive

A reflexive module is a module that is isomorphic via the natural map to its second dual.

resolution

resolution

restriction

Restriction of scalars uses a ring homomorphism from R to S to convert S-modules to R-modules.

S

Schanuel

Schanuel's lemma

Schur

Schur's lemma says that the endomorphism ring of a simple module is a division ring.

Shapiro

Shapiro's lemma

sheaf of modules

sheaf of modules

snake

snake lemma

socle

The socle is the largest semisimple submodule.

semisimple

A semisimple module is a direct sum of simple modules.

simple

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

Smith

Smith normal form

stably free

A stably free module

structure theorem

The structure theorem for finitely generated modules over a principal ideal domain says that a finitely generated modules over PIDs are finite direct sums of primary cyclic modules.

submodule

Given a [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ M }[/math], an additive subgroup [math]\displaystyle{ N }[/math] of [math]\displaystyle{ M }[/math] is a submodule if [math]\displaystyle{ RN \subset N }[/math].

support

The support of a module over a commutative ring is the set of prime ideals at which the localizations of the module are nonzero.

T

tensor

Tensor product of modules

topological

A topological module

Tor

Tor functor

torsion-free

torsion-free module

torsionless

torsionless module

U

uniform

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

W

weak

weak dimension

Z

zero

1.  The zero module is a module consisting of only zero element.

2.  The zero module homomorphism is a module homomorphism that maps every element to zero.

References

• John A. Beachy (1999). Introductory Lectures on Rings and Modules (1st ed.). Addison-Wesley. ISBN 0-521-64407-0. https://archive.org/details/introductorylect0000beac. 
• Golan, Jonathan S.; Head, Tom (1991), Modules and the structure of rings, Monographs and Textbooks in Pure and Applied Mathematics, 147, Marcel Dekker, ISBN 978-0-8247-8555-0, https://archive.org/details/modulesstructure0000gola 
• Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5 
• Serge Lang (1993). Algebra (3rd ed.). Addison-Wesley. ISBN 0-201-55540-9. 
• Passman, Donald S. (1991), A course in ring theory, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 978-0-534-13776-2