Θεωρία
Glossary of module theory, List of mathematical theories
Προσεγγισιακή Θεωρία
(Approximation theory) Ασυμπτωτική Θεωρία
(Asymptotic theory) Αυτοματική Θεωρία
(Automata theory) Διακλαδωσιακή Θεωρία
(Bifurcation theory) Πλεξιδοθεωρία (Braid theory) Καταστροφοθεωρία Θεωρία
(Catastrophe theory) Κατηγοροθεωρία (Category theory) Χαοτική Θεωρία (Chaos theory) Χαρακτηροθεωρία
(Character theory) Θεωρία (Choquet theory) Κωδικοθεωρία (Coding theory) Θεωρία (Cohomology theory) Θεωρία (Computation theory) Παραμορφωσιακή Θεωρία
(Deformation theory) Διαστασοθεωρία (Dimension theory) Κατανομική Θεωρία
(Distribution theory) Πεδιακή Θεωρία (Field theory) Εξαλειπτική Θεωρία
(Elimination theory) Γραφοθεωρία (graph theory) Θεωρία Galois (Galois theory) Παιγνιοθεωρία (Game theory) Ομαδοθεωρία (Group theory) Θεωρία Hodge (Hodge theory) Ομολογοθεωρία (Homology theory) Ομοτοποθεωρία (Homotopy theory) Πληροφορική Θεωρία
(Information theory) Αναλλοιωτική Θεωρία
(Invariant theory) Θεωρία K (K-theory) Κομβοθεωρία (Knot theory) Θεωρία L (L-theory) Θεωρία M (M-theory) Μητροθεωρία (Matrix theory) Μετροθεωρία (Measure theory) Προτυπική Θεωρία
(Model theory) Θεωρία Morse (Morse theory) Μοδιοθεωρία (Module theory) Δικτυακή Θεωρία (Network theory) Θεωρία Nevanlinna
(Nevanlinna theory) Αριθμοθεωρία (Number theory) Παρεμποδιακή Θεωρία
(Obstruction theory) Τελεστική Θεωρία (Operator theory) Διαταξιακή Θεωρία (Order theory) Διηθητική Θεωρία
(Percolation theory) Διαταρακτική Θεωρία
(Perturbation theory) Πιθανοτική Θεωρία
(Probability theory) Αποδειξιακή Θεωρία (Proof theory) Κβαντική Θεωρία (Quantum theory) Στοιχισιακή Θεωρία (Queue theory) Αναδρομοθεωρία (Recursion theory) Αναπαραστασιακή Θεωρία
(Representation theory) Δακτυλιοθεωρία (Ring theory) Σχεδιοθεωρία (Scheme theory) Θεωρία Seiberg-Witten
(Seiberg-Witten theory) Τυποθεωρία (Set theory) Δραγμοθεωρία (Sheaf theory) Μοναδικοτική Θεωρία
(Singularity theory) Φασματοθεωρία (Spectral theory) Χορδοθεωρία (String theory) Χειρουργοθεωρία (Surgery theory) Εξισωσοθεωρία
(Theory of equations) Τοποθεωρία (Topos theory) Συστροφική Θεωρία
(Twistor theory) Τυποθεωρία (Type theory)
Φυσικές Θεωρίες Χημικές ΘεωρίεςΓεωλογικές Θεωρίες Βιολογικές Θεωρίες Οικονομικές Θεωρίες Φιλοσοφικές Θεωρίες
- Μια Μαθηματική Θεωρία.
Ετυμολογία[]
Η ονομασία "μαθηματική" σχετίζεται ετυμολογικά με την λέξη "Μαθηματικά".
Εισαγωγή[]
Ορολογία[]
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.
Υποσημειώσεις[]
Εσωτερική Αρθρογραφία[]
- Επιστημονική Θεωρία
- Μαθηματική Θεωρία
- Προτυποθεωρία (Model theory)
- Διαταξιακή θεωρία (order theory)
- Συνολοθεωρία (Set theory)
- Αποδειξιακή θεωρία (Proof theory)
- Αριθμοθεωρία (Number theory)
- Δακτυλιοθεωρία (ring theory)
- Γραφοθεωρία (graph theory)
- Διαστασοθεωρία (Dimension theory)
- Ομολογιακή θεωρία (homology theory)
- ομοτοπική θεωρία (Homotopy theory)
- Πιθανοτική Θεωρία (Probability theory)
- Κομβοθεωρία (Knot theory)
- Ιδεωδοθεωρία (Ideal theory)
- Φυσική Θεωρία, Πεδιακή Θεωρία
Βιβλιογραφία[]
Ιστογραφία[]
 Κίνδυνοι Χρήσης
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