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Algebraic structure in ring theory
algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion-free modules. Formally, a module M over a
Flat_module
Direct summand of a free module (mathematics)
projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free modules. Various
Projective_module
Toy robot arm
positions on the flat module, open the lid of the box, and then remove the cylinders from the box and place them in position on the flat module. It is possible
Armatron
Exact sequence used to describe the structure of an object
projective modules or flat modules. Similarly every module has injective resolutions, which are right resolutions consisting of injective modules. Given a
Resolution_(algebra)
Module components with flexibility in module theory
module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups. While flat modules are those modules which leave
Pure_submodule
Operation that pairs a left and a right R-module into an abelian group
of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction
Tensor_product_of_modules
Mathematical object in abstract algebra
abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers
Injective_module
In algebra, module with a finite generating set
generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over
Finitely_generated_module
Topics referred to by the same term
Euclidean space Flat (matroids), a further generalization of flats from linear algebra to the context of matroids Flat module in ring theory Flat morphism in
Flat
Generalization of vector spaces from fields to rings
desirable properties. Injective Injective modules are defined dually to projective modules. Flat A module is called flat if taking the tensor product of it with
Module_(mathematics)
Topics referred to by the same term
Faithfully flat may refer to: Faithfully flat morphism, in the theory of schemes in algebraic geometry Faithfully flat module, for sequences in algebra
Faithfully_flat
algebra, a flat cover of a module M over a ring is a surjective homomorphism from a flat module F to M that is in some sense minimal. Any module over a ring
Flat_cover
sometimes called the flat dimension as it is the shortest length of the resolution of the module by flat modules. The weak dimension of a module is, at most,
Weak_dimension
In mathematics, a module that has a basis
mathematics, a free module is a module that has a basis, that is, a generating set that is linearly independent. Every vector space is a free module, but, if the
Free_module
Topics referred to by the same term
mathematical analysis Flat intonation Flat module in abstract algebra Flat morphism in algebraic geometry Flat (disambiguation) Flattening This disambiguation
Flatness
Module over a ring
module. Over a commutative ring R with total quotient ring K, a module M is torsion-free if and only if Tor1(K/R,M) vanishes. Therefore flat modules,
Torsion-free_module
theory of descent (faithfully flat descent). The term flat here comes from flat modules. There are several slightly different flat topologies, the most common
Flat_topology
of R are flat. All left ideals of R are flat. Submodules of all right flat R-modules are flat. Submodules of all left flat R-modules are flat. (The mixture
Torsionless_module
Construction of a ring of fractions
"denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions
Localization (commutative algebra)
Localization_(commutative_algebra)
module. flat 1. A flat module is a module such that tensoring with it preserves exactness. 2. A flat resolution is a resolution by flat modules. 3. For
Glossary of commutative algebra
Glossary_of_commutative_algebra
criterion for flatness gives conditions one can check to show flatness of a module. Given a commutative ring A, an ideal I and an A-module M, suppose either
Local_criterion_for_flatness
Component of the Apollo spacecraft
The Apollo command and service module (CSM) was one of two principal components of the United States Apollo spacecraft, used for the Apollo program, which
Apollo command and service module
Apollo_command_and_service_module
Topics referred to by the same term
indicators, used to operate machinery Flat panel display, in (for example) laptops and mobile devices Solar panel, a flat module of photovoltaic solar cells Panel
Panel
Homological map
the homology is of the complexes formed by tensor product with C (some flat module condition should enter). The construction of β is by the usual argument
Bockstein_homomorphism
Concept in mathematics
a module. Countably generated module Flat module Invariant basis number "ac.commutative algebra – Existence of a minimal generating set of a module –
Generating_set_of_a_module
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
Glossary_of_module_theory
Replaceable RAM form factor
the SOCAMM2 module sits flat to the main board, is approximately 90mm long, 14mm wide and around 1mm high. It has 694 pins. Each module can integrate
CAMM_(memory_module)
Assembly of photovoltaic cells used to generate electricity
device that converts sunlight into electricity by using multiple solar modules that consist of photovoltaic (PV) cells. PV cells are made of materials
Solar_panel
Flat expanse of ground covered with salt and other minerals
Natural salt flats or salt pans are flat, open expanses of ground covered with salt and other minerals, usually shining white under the sun. They are
Salt_flat
instance, being a flat module over a commutative ring is a local property, but being a free module is not. For more, see Localization of a module. Local path
Local_property
Zero divisors in a module
is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed
Torsion_(algebra)
Module generated by a countable subset
if and only if (i) it is flat, (ii) it is a direct sum of countably generated modules and (iii) it is a Mittag-Leffler module. (Bazzoni–Stovicek) Kaplansky
Countably_generated_module
Branch of mathematics that studies algebraic structures
Injective module Injective hull Flat module Flat cover Coherent module Finitely-generated module Finitely-presented module Finitely related module Algebraically
List of abstract algebra topics
List_of_abstract_algebra_topics
In algebra, completion w.r.t. powers of an ideal
any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and
Completion_of_a_ring
Guidance and navigation computer used in Apollo spacecraft
program that was installed on board each Apollo command module (CM) and Apollo Lunar Module (LM). The AGC provided computation and electronic interfaces
Apollo_Guidance_Computer
Algebraic structure
general R-module M, the functor M ⊗R − is only right exact. If it is exact, M is called flat. If R is local, any finitely presented flat module is free
Commutative_ring
Notion in abstract algebra
of a projective cover does not always exist for a module, however a flat cover exists for every module. In some cases, for R {\displaystyle R} a subring
Injective_hull
Technique from algebraic geometry
A\to B} , the faithfully flat descent is, roughly, the statement that to give a module or an algebra over A is to give a module or an algebra over B {\displaystyle
Faithfully_flat_descent
Low-cost, low-power SoC microcontrollers with Bluetooth and Wi-Fi
power amplifiers, low-noise receivers, filters, and power-management modules. Typically, the ESP32 is embedded on device-specific printed circuit boards
ESP32
results discovered by Joachim Lambek shows that a module is flat if and only if the associated character module is injective. The group ( Q / Z , + ) {\displaystyle
Character_module
Commutative algebra studies commutative rings, their ideals, and modules over such rings
an ideal Hilbert's Nullstellensatz Flat module Flat map Flat map (ring theory) Projective module Injective module Cohen-Macaulay ring Gorenstein ring
List of commutative algebra topics
List_of_commutative_algebra_topics
Object file format
support the 80386 32-bit flat address mode. […] The OMF gave us a good roadmap for the design of the linker, which knitted object modules together, and the locator
Object_Module_Format_(Intel)
Module over the non-commutative Dieudonné ring
In mathematics, a Dieudonné module introduced by Jean Dieudonné (1954, 1957b), is a module over the non-commutative Dieudonné ring, which is generated
Dieudonné_module
Two closely related mathematical subjects
had not yet been invented by the time of GAGA's publication. Flat module - Notion of flatness was introduced by Serre (1956). Algebraic and analytic local
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
Category with direct sums and certain types of kernels and cokernels
category, flatness refers to the idea that − ⊗ A {\displaystyle -\otimes A} is an exact functor. See flat module or, for more generality, flat morphism
Abelian_category
Electronic toy made by Texas Instruments
and a receptor slot to receive one of a collection of ROM game library modules. The first Speak & Spell was introduced at the summer Consumer Electronics
Speak_&_Spell_(toy)
Mathematical terminology
In mathematics, a Galois module is a G-module, with G being the Galois group (named for Évariste Galois) of some extension of fields. The term Galois representation
Galois_representation
Method to produce electricity from solar radiation
silicon solar cell prices were at $76.67/W. Although wholesale module prices remained flat at around $3.50 to $4.00/W in the early 2000s due to high demand
Photovoltaics
the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due
Generic_flatness
Slot-in computer standard
Pluggable Specification (OPS) is a computing module plug-in format available for adding computing capability to flat panel displays. The format was first announced
Open_Pluggable_Specification
Docking compartment of the International Space Station
exterior of Zvezda's Service Module. On 10 June 2009, during Russian Orlan EVA-23 spacewalk Padalka and Barratt replaced a flat hatch cover in the forward
Poisk_(ISS_module)
First crewed Moon landing (1969)
mission was crewed by Commander Neil Armstrong, Command Module Pilot Michael Collins, and Lunar Module Pilot Edwin "Buzz" Aldrin, all of whom were on their
Apollo_11
Module over a sheaf of differential operators
In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of
D-module
Local ring in which Hensel's lemma holds
Henselization of A has the same completion and residue field as A and is a flat module over A. If A is Noetherian, reduced, normal, regular, or excellent then
Henselian_ring
Overview of and topical guide to algebraic structures
values in F. Other special types of modules, including free modules, projective modules, injective modules and flat modules are studied in abstract algebra
Outline of algebraic structures
Outline_of_algebraic_structures
Direct sum of irreducible modules
area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from
Semisimple_module
Multi-stage paper folding technique
folding technique in which individual modules or units are created out of sheets of paper and assembled into a flat shape or three-dimensional structure
Modular_origami
Scheme theory concept
P}} is a flat map for all P in X. A map of rings A → B {\displaystyle A\to B} is called flat if it is a homomorphism that makes B a flat A-module. A morphism
Flat_morphism
Operation in algebra
refers to one of several related ways for changing the coefficient ring of a module. Given (possibly non-commutative) unital rings R {\displaystyle R} and S
Change_of_rings
Second crewed Moon landing
visible from Earth; they had to be relatively flat and without major obstructions along the path the Lunar Module (LM) would fly to reach them, their suitability
Apollo_12
Having curvature zero. 3. (Adjective) For the term "flat" in scheme theory see flat module, flat morphism. flecnode A double point that is also a point
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Web page delivered to web browser as-is
A static web page, sometimes called a flat page or a stationary page, is a web page that is delivered to a web browser exactly as stored, in contrast to
Static_web_page
Multinational wheeled armoured fighting vehicle
drive module and interchangeable mission modules which allow several configurations to meet different operational requirements. The drive module has been
Boxer (armoured fighting vehicle)
Boxer_(armoured_fighting_vehicle)
Rings admitting weak inverses
are also called absolutely flat rings, because these rings are characterized by the fact that every left R-module is flat. Von Neumann regular rings were
Von_Neumann_regular_ring
Generalization of vector bundles
finitely generated projective modules over R {\displaystyle R} , or (equivalently) to finitely generated flat modules over R {\displaystyle R} . Let
Coherent_sheaf
Modular origami module
tetrahedron, using six units, results in a cube (the central fold of each module lays flat, creating square faces instead of isosceles right triangular faces
Sonobe
Series of mathematics books by Nicolas Bourbaki
Algèbre commutative: Chapitres 1 à 4 1 Modules plats Commutative Algebra Commutative Algebra: Chapters 1-7 1 Flat Modules 2 Localisation 2 Localization 3 Graduations
Éléments_de_mathématique
Modular handheld computer
device when it is on a flat surface. Inside, the components are modular, with a ClockworkPi mainboard which takes a compute module (or the CM4 adapter to
UConsole
injective envelopes and flat covers, which exist for every left (right) R-module regardless of the ring R, left (right) R-modules do not in general have
Projective_cover
Family of computer storage devices
Storage Module Interfaces with Extensions for Enhanced Storage Module Interfaces. The SMD interface is based upon a definition of two flat interface
Storage_Module_Device
Equivalence relation on rings
S-Mod, then the R module M has any of the following properties if and only if the S module F(M) does: injective, projective, flat, faithful, simple,
Morita_equivalence
Polymer frame semi-automatic handgun
unit (FCU) as the P365, while having a larger grip module that accepts 12-round flush magazines, a flat trigger, and a longer slide and barrel. The rear
SIG_Sauer_P365
Fifth crewed Moon landing
The mission was crewed by Commander John Young, Lunar Module Pilot Charles Duke and Command Module Pilot Ken Mattingly. Launched from the Kennedy Space
Apollo_16
Quillen's original short proof uses generic flatness. Quillen, D. (1969). "On the endomorphism ring of a simple module over an enveloping algebra". Proceedings
Quillen's_lemma
Main body and structural component of the satellite
Life support (for crewed missions). Comparison of satellite buses Service module Satellite "TU Delft: Spacecraft bus subsystems". Lr.tudelft.nl. Archived
Satellite_bus
Type of computer memory introduced 2014
As a prototype, this DDR4 memory module has a flat edge connector at the bottom, while production DDR4 DIMM modules have a slightly curved edge connector
DDR4_SDRAM
of R. Flatness Every torsion-free R-module is flat. Every torsionless R-module is flat. Every ideal of R is flat. Every overring of R is R-flat. Every
Prüfer_domain
Construction in homological algebra
B {\displaystyle B} is flat (for example, free) as an R {\displaystyle R} -module. In fact, one can compute Tor using a flat resolution of either A {\displaystyle
Tor_functor
Index of articles associated with the same name
of a module, based on projective resolutions Injective dimension of a module, based on injective resolutions Weak dimension of a module, or flat dimension
Homological_dimension
ring being left perfect.) Every flat left R-module is projective. R/J(R) is semisimple and every non-zero left R-module contains a maximal submodule. R
Perfect_ring
Mathematical space with two coordinates
plane or complex plane. The most basic example is the flat Euclidean plane, an idealization of a flat surface in physical space such as a sheet of paper
Two-dimensional_space
Type of commutative ring in mathematics
Cohen–Macaulay if and only if it is flat as an A-module; it is also equivalent to say that R is free as an A-module. A geometric reformulation is as follows
Cohen–Macaulay_ring
Telecommunications feature
telephone call, typically limited by settings in the autopatch module to be only to flat-rate numbers, such as local calls or toll-free numbers. The practice
Autopatch
Area that contained a standing surface water body
alkali flat. If covered with salt, it is called a salt flat. If its basin is primarily salt, then a dry lake bed is called a salt pan, pan, or salt flat (the
Dry_lake
of a module M at an ideal I (often a maximal ideal) e I ( M ) . {\displaystyle \mathbf {e} _{I}(M).} The notion of the multiplicity of a module is a generalization
Multiplicity_theory
Database stored as flat data
A flat-file database is a tabular flat file in which each record is semantically independent – can meaningfully be interpreted and manipulated independent
Flat-file_database
Doughnut-shaped surface of revolution
{\displaystyle T^{n}} , Z) can be identified with the exterior algebra over the Z-module Zn whose generators are the duals of the n nontrivial cycles. As the n-torus
Torus
Spanish scientist
that used parquets of very compact non-imaging lenses assembled into flat modules such as the conventional PV ones. In 2007 he co-founded and was the chairman
Antonio_Luque
British sports car developed by McLaren Automotive
powertrain, combining a 4.0-liter twin-turbo flat-plane crank V8 engine (designated MHP-8) with an electric module that includes a radial flux electric motor
McLaren_W1
Oat bar made with butter, sugar & syrup
known as a cereal bar, oat bar or oat slice) is a baked bar, cooked in a flat oven tin and cut into squares or rectangles, made from rolled oats, fat (typically
Flapjack_(oat_bar)
Flat-panel displays are thin panels of glass or plastic used for electronically displaying text, images, or video. Liquid crystal displays (LCD), OLED
List of flat panel display manufacturers
List_of_flat_panel_display_manufacturers
Standard-width frame for electronic equipment
standardized frame or enclosure for mounting multiple electronic equipment modules. Each module has a front panel that is 19 inches (482.6 mm) wide. The 19-inch
19-inch_rack
Musical terminology
An altered chord is a chord that replaces one or more notes from the diatonic scale with a neighboring pitch from the chromatic scale. By the broadest
Altered_chord
IBM hybrid circuit technology introduced in 1964
an SLT module. The circuits were either encapsulated in plastic (early prototypes) or covered with a metal lid. Several of these SLT modules (20 in the
Solid_Logic_Technology
On when a morphism of spectral sequences in homological algebra is an isomorphism
q}^{r},{}^{\prime }E_{p,q}^{r}} be first quadrant spectral sequences of flat modules over a commutative ring and f : E r → ′ E r {\displaystyle f:E^{r}\to
Zeeman's_comparison_theorem
Local and global geometry of the universe
3 torus has everywhere zero curvature but is finite in extent, whereas a flat simply connected space is infinite in extent (such as Euclidean space). Observational
Shape_of_the_universe
Facet of general relativity
total mass exist in important classes of spacetimes, such as asymptotically flat spacetimes and asymptotically anti-de Sitter spacetimes, where suitable boundary
Mass_in_general_relativity
2026 foldable smartphone developed by Google
extend all the way to the right edge, covering over the microphone and flash module also. On the top of the two stacked camera covers, the flash moves over
Pixel_11_Pro_Fold
US Navy fast attack submarine class
section form a module which, when joined with other modules, forms a Virginia-class submarine. Block I boats were built in 10 modules with each submarine
Virginia-class_submarine
Objects between rings and their fields of fractions
m ⊗ m is a flat V-module. A module N over V is almost zero with respect to such m if for all ε ∈ m and n ∈ N we have εn = 0. Almost zero modules form a Serre
Almost_ring
travel, tourism, insurance
FLAT MODULE
FLAT MODULE
FLAT MODULE
FLAT MODULE
FLAT MODULE
FLAT MODULE
FLAT MODULE
FLAT MODULE
FLAT MODULE
travel, tourism, insurance