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 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 -module is the set . It is a (left) ideal of .
2.  The annihilator of an element is the set .
Artinian
An Artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.
associated prime
1.  An associated prime.
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 is a set of elements in 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
bimdule
bimodule

C

character
character module
coherent
A coherent module is a finitely generated module whose finitely generated submodules are finitely presented.
completely reducible
Synonymous to "semisimple module".
composition
Jordan Hölder composition series
continuous
continuous module
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.
dense
dense submodule
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 .
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
The endomorphism ring.
essential
Given a module M, an essential submodule N of M is a submodule that every nonzero submodule of M intersects non-trivially.
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 is one where the action of each nonzero on is nontrivial (i.e. for some in ). Equivalently, 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 where are finitely generated free modules.
2.  A finitely presented module is a module that admits a finite free presentation.
finitely generated
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 .
fitting
fitting ideal
five
Five lemma.
flat
A -module is called a flat module if the tensor product functor 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 .

G

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

H

graded
A module over a graded ring is a graded module if can be expressed as a direct sum and .
homomorphism
For two left -modules , a group homomorphism is called homomorphism of -modules if .
Hom
Hom functor.

I

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).
injective
1.  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 module Q is injective if the diagram commutes
The following conditions are equivalent:
  • The contravariant functor is exact.
  • is a injective module.
  • Every short exact sequence is split.
2.  An injective envelope 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.

J

Jacobson
density theorem

K

Kaplansky
Kaplansky's theorem on a projective module says that a projective module over a local ring is free.
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.
localization
Localization of a module converts R modules to S modules, where S is a localization of R.

M

Mitchell's embedding theorem
Mitchell's embedding theorem
Mittag-Leffler
Mittag-Leffler condition (ML)
module
1.  A left module over the ring is an abelian group with an operation (called scalar multipliction) satisfies the following condition:
,
2.  A right module over the ring is an abelian group with an operation satisfies the following condition:
,
3.  All the modules together with all the module homomorphisms between them form the category of modules.

N

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

principal
A principal indecomposable module is a cyclic indecomposable projective module.
primary
A primary submodule
projective
The characteristic property of projective modules is called lifting.
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.
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.

Q

quotient
Given a left -module and a submodule , the quotient group can be made to be a left -module by for . 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
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.
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 -module , an additive subgroup of is a submodule if .
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
Tor
Tor functor.
torsionless
A torsionless module.

U

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

References

  • John A. Beachy (1999). Introductory Lectures on Rings and Modules (1st ed.). Addison-Wesley. ISBN 0-521-64407-0.
  • 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, MR 1201818
  • 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, MR 1653294
  • 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, MR 1096302
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