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: The annihilator of a left R-module M is the set. It is a (left) ideal of R. The annihilator of an element m \in M 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: 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. 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, 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 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 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 is a graded module if M can be expressed as a direct sum and.

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 is called homomorphism of R-modules if. 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, 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. 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: Jacobson's 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 (called scalar multipliction) satisfies the following condition: A right module M over the ring R is an abelian group (M, +) with an operation satisfies the following condition: 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. 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 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 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. 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.