Glossary of module theory

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: ~ rm = 0 \, \forall m \in M \}$. It is a (left) ideal of $R$. The annihilator of an element $m \in M$ is the set $\textrm{Ann}(m) := \{ r \in R ~

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

associated prime: 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 $M$ is a set of elements in $M$ 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 $\operatorname{Hom}_R(M, R)$.

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: An endomorphism is a module homomorphism from a module to itself. 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 $M$ is one where the action of each nonzero $r \in R$ on $M$ is nontrivial (i.e. $rx \ne 0$ for some $x$ in $M$). Equivalently, $\textrm{Ann}(M)$ 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: A finite free presentation of a module M is an exact sequence $F_1 \to F_0 \to M$ where $F_i$ are finitely generated free modules. A finitely presented module is a module that admits a finite free presentation.

finitely generated: A module $M$ is finitely generated if there exist finitely many elements $x_1,...,x_n$ in $M$ such that every element of $M$ is a finite linear combination of those elements with coefficients from the scalar ring $R$.

fitting: fitting ideal Fitting's lemma

five: Five lemma

flat: A $R$-module $F$ is called a flat module if the tensor product functor $- \otimes_R F$ 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 $R$.

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

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

Hilbert: Hilbert's syzygy theorem The Hilbert–Poincaré series of a graded module. The Hilbert–Serre theorem tells when a Hilbert–Poincaré series is a rational function.

homological dimension: homological dimension

homomorphism: For two left $R$-modules $M_1, M_2$, a group homomorphism $\phi: M_1 \to M_2$ is called homomorphism of $R$-modules if $r \phi(m) = \phi (r m) \, \forall r \in R, m \in M_1$.

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 $f : M \to M$ is the difference $\operatorname{length}(\operatorname{coker}(f)) - \operatorname{length}(\operatorname{ker}(f))$, when the cokernel and kernel of $f$ have finite length.

injective: A $R$-module $Q$ is called an injective module if given a $R$-module homomorphism $g: X \to Q$, and an injective $R$-module homomorphism $f: X \to Y$, there exists a $R$-module homomorphism $h : Y \to Q$ such that $f \circ h = g$.
 * The following conditions are equivalent:
 * The contravariant functor $\textrm{Hom}_R( -, I)$ is exact.
 * $I$ is a injective module.
 * Every short exact sequence $0 \to I \to L \to L' \to 0$ is split.

An injective envelope (also called injective hull) is a maximal essential extension, or a minimal embedding in an injective module. 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: A linear map is another term for a module homomorphism. 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: A left module $M$ over the ring $R$ is an abelian group $(M, +)$ with an operation $R \times M \to M$ (called scalar multipliction) satisfies the following condition:
 * $\forall r,s \in R, \forall m,n \in M$,
 * $r (m + n) = rm + rn$
 * $r (s m) = (r s) m$
 * $1_R \, m = m$

A right module $M$ over the ring $R$ is an abelian group $(M, +)$ with an operation $M \times R \to M$ satisfies the following condition:
 * $\forall r,s \in R, \forall m,n \in M$,
 * $(m + n) r = m r + n r$
 * $(m s) r = r (s m)$
 * $m 1_R = m$

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: perfect complex perfect module

principal: A principal indecomposable module is a cyclic indecomposable projective module.

primary: primary submodule

projective: Projective module.png'''.]]A $R$-module $P$ is called a projective module if given a $R$-module homomorphism $g: P \to M$, and a surjective $R$-module homomorphism $f: N \to M$, there exists a $R$-module homomorphism $h : P \to N$ such that $f \circ h = g$.
 * The following conditions are equivalent:
 * The covariant functor $\textrm{Hom}_R(P, - )$ is exact.
 * $M$ is a projective module.
 * Every short exact sequence $0 \to L \to L' \to P \to 0$ is split.
 * $M$ is a direct summand of free modules.
 * In particular, every free module is projective.

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. 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 $R$-module $M$ and a submodule $N$, the quotient group $M/N$ can be made to be a left $R$-module by $r(m+N) = rm + N $ for $r \in R, \, m \in M$. 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 $R$-module $M$, an additive subgroup $N$ of $M$ is a submodule if $RN \subset N$.

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: The zero module is a module consisting of only zero element. The zero module homomorphism is a module homomorphism that maps every element to zero.