glossary of module theory

{{glossary}}

{{term|algebraically compact}}

{{defn|1=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.}}

{{term|annihilator}}

{{defn|no=1|1=The annihilator of a left $R$-module $M$ is the set $\backslash textrm\{Ann\}(M)\; :=\; \backslash \{\; r\; \backslash in\; R\; ~|~\; rm\; =\; 0\; \backslash ,\; \backslash forall\; m\; \backslash in\; M\; \backslash \}$ . It is a (left) ideal of $R$.}}

{{defn|no=2|The annihilator of an element $m\; \backslash in\; M$ is the set $\backslash textrm\{Ann\}(m)\; :=\; \backslash \{\; r\; \backslash in\; R\; ~|~\; rm\; =\; 0\; \backslash \}$.}}

{{term|Artinian}}

{{defn|1=An Artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.}}

{{term|associated prime}}

{{defn|no=1|associated prime}}

{{term|1=automorphism}}

{{defn|1=An automorphism is an endomorphism that is also an isomorphism.}}

{{term|1=Azumaya}}

{{defn|1=Azumaya's theorem says that two decompositions into modules with local endomorphism rings are equivalent.}}

{{glossary end}}

{{glossary}}

{{term|balanced}}

{{defn|1=balanced module}}

{{term|basis}}

{{defn|1=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.}}

{{term|Beauville–Laszlo}}

{{defn|Beauville–Laszlo theorem}}

{{term|big}}

{{defn|1="big" usually means "not-necessarily finitely generated".}}

{{term|bimodule}}

{{defn|1=bimodule}}

{{glossary end}}

{{glossary}}

{{term|canonical module}}

{{defn|canonical module (the term "canonical" comes from canonical divisor)}}

{{term|category}}

{{defn|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.}}

{{term|character}}

{{defn|character module}}

{{term|chain complex}}

{{defn|chain complex (frequently just complex)}}

{{term|closed submodule}}

{{defn|1=A module is called a closed submodule if it does not contain any essential extension.}}

{{term|Cohen–Macaulay}}

{{defn|Cohen–Macaulay module}}

{{term|coherent}}

{{defn|1=A coherent module is a finitely generated module whose finitely generated submodules are finitely presented.}}

{{term|cokernel}}

{{defn|1=The cokernel of a module homomorphism is the codomain quotiented by the image.}}

{{term|compact}}

{{defn|1=A compact module}}

{{term|completely reducible}}

{{defn|1=Synonymous to "semisimple module".}}

{{term|completion}}

{{defn|1=completion of a module}}

{{term|composition}}

{{defn|1=Jordan Hölder composition series}}

{{term|continuous}}

{{defn|1=continuous module}}

{{term|countably generated}}

{{defn|1=A countably generated module is a module that admits a generating set whose cardinality is at most countable.}}

{{term|cyclic}}

{{defn|1=A module is called a cyclic module if it is generated by one element.}}

{{glossary end}}

{{glossary}}

{{term|D}}

{{defn|1=A D-module is a module over a ring of differential operators.}}

{{term|decomposition}}

{{defn|1=A decomposition of a module is a way to express a module as a direct sum of submodules.}}

{{term|dense}}

{{defn|1=dense submodule}}

{{term|determinant}}

{{defn|1=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.}}

{{term|differential}}

{{defn|1=A differential graded module or dg-module is a graded module with a differential.}}

{{term|direct sum}}

{{defn|1=A direct sum of modules is a module that is the direct sum of the underlying abelian group together with component-wise scalar multiplication.}}

{{term|dual module}}

{{defn|1=The dual module of a module M over a commutative ring R is the module $\backslash operatorname\{Hom\}\_R(M,\; R)$.}}

{{term|dualizing}}

{{defn|1=dualizing module}}

{{term|Drinfeld}}

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

{{glossary end}}

{{glossary}}

{{term|Eilenberg–Mazur}}

{{defn|Eilenberg–Mazur swindle}}

{{term|elementary}}

{{defn|elementary divisor}}

{{term|endomorphism}}

{{defn|no=1|An endomorphism is a module homomorphism from a module to itself.}}

{{defn|no=2|The endomorphism ring is the set of all module homomorphisms with addition as addition of functions and multiplication composition of functions.}}

{{term|enough}}

{{defn|enough injectives}}

{{defn|enough projectives}}

{{term|essential}}

{{defn|1=Given a module M, an essential submodule N of M is a submodule that every nonzero submodule of M intersects non-trivially.}}

{{term|exact}}

{{defn|1=exact sequence}}

{{term|Ext functor}}

{{defn|1=Ext functor}}

{{term|extension}}

{{defn|1=Extension of scalars uses a ring homomorphism from R to S to convert R-modules to S-modules.}}

{{glossary end}}

{{glossary}}

{{term|faithful}}

{{defn|1=A faithful module $M$ is one where the action of each nonzero $r\; \backslash in\; R$ on $M$ is nontrivial (i.e. $rx\; \backslash ne\; 0$ for some $x$ in $M$). Equivalently, $\backslash textrm\{Ann\}(M)$ is the zero ideal.}}

{{term|finite}}

{{defn|1=The term "finite module" is another name for a finitely generated module.}}

{{term|finite length}}

{{defn|1=A module of finite length is a module that admits a (finite) composition series.}}

{{term|finite presentation}}

{{defn|no=1|1=A finite free presentation of a module M is an exact sequence $F\_1\; \backslash to\; F\_0\; \backslash to\; M$ where $F\_i$ are finitely generated free modules.}}

{{defn|no=2|1=A finitely presented module is a module that admits a finite free presentation.}}

{{term|1=finitely generated}}

{{defn|1=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 $R$.}}

{{term|fitting}}

{{defn|no=1|fitting ideal}}

{{defn|no=2|Fitting's lemma}}

{{term|five}}

{{defn|Five lemma}}

{{term|flat}}

{{defn|1=A $R$-module $F$ is called a flat module if the tensor product functor $-\; \backslash otimes\_R\; F$ is exact.{{br}}In particular, every projective module is flat.}}

{{term|1=free}}

{{defn|1=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 $R$.}}

{{term|1=Frobenius reciprocity}}

{{defn|1=Frobenius reciprocity.}}

{{glossary end}}

{{glossary}}

{{term|Galois}}

{{defn|1=A Galois module is a module over the group ring of a Galois group.}}

{{term|generating set}}

{{defn|1=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.}}

{{term|global}}

{{defn|1=global dimension}}

{{term|1=graded}}

{{defn|1=A module $M$ over a graded ring $A\; =\; \backslash bigoplus\_\{n\backslash in\; \backslash mathbb\; N\}A\_n$ is a graded module if $M$ can be expressed as a direct sum $\backslash bigoplus\_\{i\backslash in\; \backslash mathbb\; N\}M\_i$ and $A\_i\; M\_j\; \backslash subseteq\; M\_\{i+j\}$.}}

{{glossary end}}

{{glossary}}

{{term|1=Herbrand quotient}}

{{defn|1=A Herbrand quotient of a module homomorphism is another term for index.}}

{{term|1=Hilbert}}

{{defn|no=1|1=Hilbert's syzygy theorem}}

{{defn|no=2|1=The Hilbert–Poincaré series of a graded module.}}

{{defn|no=3|1=The Hilbert–Serre theorem tells when a Hilbert–Poincaré series is a rational function.}}

{{term|1=homological dimension}}

{{defn|1=homological dimension}}

{{term|1=homomorphism}}

{{defn|1=For two left $R$-modules $M\_1,\; M\_2$, a group homomorphism $\backslash phi:\; M\_1\; \backslash to\; M\_2$ is called homomorphism of $R$-modules if $r\; \backslash phi(m)\; =\; \backslash phi\; (r\; m)\; \backslash ,\; \backslash forall\; r\; \backslash in\; R,\; m\; \backslash in\; M\_1$ .

}}

{{term|Hom}}

{{defn|1=Hom functor}}

{{glossary end}}

{{glossary}}

{{term|idempotent}}

{{defn|1=An idempotent is an endomorphism whose square is itself.}}

{{term|indecomposable}}

{{defn|1=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).}}

{{term|index}}

{{defn|1=The index of an endomorphism $f\; :\; M\; \backslash to\; M$ is the difference $\backslash operatorname\{length\}(\backslash operatorname\{coker\}(f))\; -\; \backslash operatorname\{length\}(\backslash operatorname\{ker\}(f))$, when the cokernel and kernel of $f$ have finite length.}}

{{term|injective}}

{{defn|no=1|1=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 $f:\; X\; \backslash to\; Y$, there exists a

$R$-module homomorphism $h\; :\; Y\; \backslash to\; Q$ such that $f\; \backslash circ\; h\; =\; g$.

Image:Injective module.svg

: The following conditions are equivalent:

:* The contravariant functor $\backslash textrm\{Hom\}\_R(\; -\; ,\; I)$ is exact.

:* $I$ is a injective module.

:* Every short exact sequence $0\; \backslash to\; I\; \backslash to\; L\; \backslash to\; L\text{'}\; \backslash to\; 0$ is split.}}

{{defn|no=2|1=An injective envelope (also called injective hull) is a maximal essential extension, or a minimal embedding in an injective module.}}

{{defn|no=3|1=An injective cogenerator is an injective module such that every module has a nonzero homomorphism into it.}}

{{term|invariant}}

{{defn|invariants}}

{{term|invertible}}

{{defn|An invertible module over a commutative ring is a rank-one finite projective module.}}

{{term|irreducible module}}

{{defn|1=Another name for a simple module.}}

{{term|isomorphism}}

{{defn|1=An isomorphism between modules is an invertible module homomorphism.}}

{{glossary end}}

{{glossary}}

{{term|Jacobson}}

{{defn|1=Jacobson's density theorem}}

{{glossary end}}

{{glossary}}

{{term|Kähler differentials}}

{{defn|Kähler differentials}}

{{term|Kaplansky}}

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

{{term|kernel}}

{{defn|The kernel of a module homomorphism is the pre-image of the zero element.}}

{{term|Koszul complex}}

{{defn|Koszul complex}}

{{term|Krull–Schmidt}}

{{defn|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.}}

{{glossary end}}

{{glossary}}

{{term|1=length}}

{{defn|1=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.}}

{{term|1=linear}}

{{defn|no=1|1=A linear map is another term for a module homomorphism.}}

{{defn|no=2|1=Linear topology}}

{{term|localization}}

{{defn|Localization of a module converts R modules to S modules, where S is a localization of R.}}

{{glossary end}}

{{glossary}}

{{term|Matlis module}}

{{defn|Matlis module}}

{{term|Mitchell's embedding theorem}}

{{defn|Mitchell's embedding theorem}}

{{term|Mittag-Leffler}}

{{defn|Mittag-Leffler condition (ML)}}

{{term|1=module}}

{{defn|no=1|1=A left module $M$ over the ring $R$ is an abelian group $(M,\; +)$ with an operation $R\; \backslash times\; M\; \backslash to\; M$ (called scalar multipliction) satisfies the following condition:

:: $\backslash forall\; r,s\; \backslash in\; R,\; \backslash forall\; m,n\; \backslash in\; M$,

:# $r\; (m\; +\; n)\; =\; rm\; +\; rn$

:# $r\; (s\; m)\; =\; (r\; s)\; m$

:# $1\_R\; \backslash ,\; m\; =\; m$}}

{{defn|no=2|1=A right module $M$ over the ring $R$ is an abelian group $(M,\; +)$ with an operation $M\; \backslash times\; R\; \backslash to\; M$ satisfies the following condition:

:: $\backslash forall\; r,s\; \backslash in\; R,\; \backslash forall\; m,n\; \backslash in\; M$,

:# $(m\; +\; n)\; r\; =\; m\; r\; +\; n\; r$

:# $(m\; s)\; r\; =\; r\; (s\; m)$

:# $m\; 1\_R\; =\; m$}}

{{defn|no=3|All the modules together with all the module homomorphisms between them form the category of modules.}}

{{term|1=module spectrum}}

{{defn|1=A module spectrum is a spectrum with an action of a ring spectrum.}}

{{glossary end}}

{{glossary}}

{{term|nilpotent}}

{{defn|1=A nilpotent endomorphism is an endomorphism, some power of which is zero.}}

{{term|Noetherian}}

{{defn|1=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.}}

{{term|normal}}

{{defn|1=normal forms for matrices}}

{{glossary end}}

{{glossary}}

{{term|perfect}}

{{defn|no=1|1=perfect complex}}

{{defn|no=2|1=perfect module}}

{{term|principal}}

{{defn|1=A principal indecomposable module is a cyclic indecomposable projective module.}}

{{term|primary}}

{{defn|1=primary submodule}}

{{term|projective}}

{{defn|1=File:Projective module.pnging.]]A $R$-module $P$ is called a projective module if given a $R$-module homomorphism $g:\; P\; \backslash to\; M$, and a surjective $R$-module homomorphism $f:\; N\; \backslash to\; M$, there exists a $R$-module homomorphism $h\; :\; P\; \backslash to\; N$ such that $f\; \backslash circ\; h\; =\; g$.

: The following conditions are equivalent:

:* The covariant functor $\backslash textrm\{Hom\}\_R(P,\; -\; )$ is exact.

:* $M$ is a projective module.

:* Every short exact sequence $0\; \backslash to\; L\; \backslash to\; L\text{'}\; \backslash to\; P\; \backslash to\; 0$ is split.

:* $M$ is a direct summand of free modules.

: In particular, every free module is projective.}}

{{defn|no=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.}}

{{defn|no=3|A projective cover is a minimal surjection from a projective module.}}

{{term|pure submodule}}

{{defn|1=pure submodule}}

{{glossary end}}

{{glossary}}

{{term|Quillen–Suslin theorem}}

{{defn|1=The Quillen–Suslin theorem states that a finite projective module over a polynomial ring is free.}}

{{term|quotient}}

{{defn|1=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 $r\; \backslash in\; R,\; \backslash ,\; m\; \backslash in\; M$. It is called a quotient module or factor module.}}

{{glossary end}}

{{glossary}}

{{term|radical}}

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

{{term|rational}}

{{defn|1=rational canonical form}}

{{term|reflexive}}

{{defn|1=A reflexive module is a module that is isomorphic via the natural map to its second dual.}}

{{term|resolution}}

{{defn|1=resolution}}

{{term|restriction}}

{{defn|Restriction of scalars uses a ring homomorphism from R to S to convert S-modules to R-modules.}}

{{glossary end}}

{{glossary}}

{{term|Schanuel}}

{{defn|1=Schanuel's lemma}}

{{term|Schur}}

{{defn|1=Schur's lemma says that the endomorphism ring of a simple module is a division ring.}}

{{term|Shapiro}}

{{defn|1=Shapiro's lemma}}

{{term|sheaf of modules}}

{{defn|1=sheaf of modules}}

{{term|snake}}

{{defn|1=snake lemma}}

{{term|socle}}

{{defn|1=The socle is the largest semisimple submodule.}}

{{term|semisimple}}

{{defn|1=A semisimple module is a direct sum of simple modules.}}

{{term|simple}}

{{defn|1=A simple module is a nonzero module whose only submodules are zero and itself.}}

{{term|Smith}}

{{defn|1=Smith normal form}}

{{term|stably free}}

{{defn|1=A stably free module}}

{{term|structure theorem}}

{{defn|1=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.}}

{{term|submodule}}

{{defn|1=Given a $R$-module $M$, an additive subgroup $N$ of $M$ is a submodule if $RN\; \backslash subset\; N$.}}

{{term|support}}

{{defn|1=The support of a module over a commutative ring is the set of prime ideals at which the localizations of the module are nonzero.}}

{{glossary end}}

{{glossary}}

{{term|tensor}}

{{defn|1=Tensor product of modules}}

{{term|topological}}

{{defn|1=A topological module}}

{{term|Tor}}

{{defn|1=Tor functor}}

{{term|torsion-free}}

{{defn|1=torsion-free module}}

{{term|torsionless}}

{{defn|1=torsionless module}}

{{glossary end}}

{{glossary}}

{{term|uniform}}

{{defn|1=A uniform module is a module in which every two non-zero submodules have a non-zero intersection.}}

{{glossary end}}

{{glossary}}

{{term|weak}}

{{defn|weak dimension}}

{{glossary end}}

{{anchor|XYZ}}

{{glossary}}

{{term|zero}}

{{defn|no=1|1=The zero module is a module consisting of only zero element.}}

{{defn|no=2|1=The zero module homomorphism is a module homomorphism that maps every element to zero.}}

{{glossary end}}

• {{cite book | author=John A. Beachy | title=Introductory Lectures on Rings and Modules | edition=1st | publisher=Addison-Wesley | year=1999 | isbn=0-521-64407-0 | url-access=registration | url=https://archive.org/details/introductorylect0000beac }}
• {{citation | last1=Golan | first1=Jonathan S. | last2=Head | first2=Tom | title=Modules and the structure of rings | publisher=Marcel Dekker | series=Monographs and Textbooks in Pure and Applied Mathematics | isbn=978-0-8247-8555-0 | mr=1201818 | year=1991 | volume=147 | url-access=registration | url=https://archive.org/details/modulesstructure0000gola }}
• {{citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 |mr=1653294 | year=1999}}
• {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd | publisher=Addison-Wesley | year=1993 | isbn=0-201-55540-9 }}
• {{citation | last1=Passman | first1=Donald S. | title=A course in ring theory | publisher=Wadsworth & Brooks/Cole Advanced Books & Software | location=Pacific Grove, CA | series=The Wadsworth & Brooks/Cole Mathematics Series | isbn=978-0-534-13776-2 |mr=1096302 | year=1991}}

Module

Category:Wikipedia glossaries using description lists