[permalink] [id link]
Algebraically compact modules share many other properties with injective objects because of the following: there exists an embedding of R-Mod into a Grothendieck category G under which the algebraically compact R-modules precisely correspond to the injective objects in G.
from
Wikipedia
Some Related Sentences
Algebraically and with
; Algebraically independent elements: Elements of an extension field of F are algebraically independent over F if they don't satisfy any non-zero polynomial equation with coefficients in F.
Algebraically, a map is anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar times a reflection at every point.
Algebraically, a lattice is a set with two associative, commutative idempotent binary operations linked by corresponding absorption laws.
Algebraically it is generated by applying the rules of vectors and coordinate geometry to a space with four dimensions.
Algebraically and into
Algebraically, let a + b be the circumference of a circle, divided into a longer arc of length a and a smaller arc of length b such that
Algebraically and algebraically
; Algebraically closed field: A field F is algebraically closed if every polynomial in F has a root in F ; equivalently: every polynomial in F is a product of linear factors.
Algebraically and .
Algebraically, these units can be thought of as the squares of the corresponding length units.
Algebraically, this means that the standard MV-algebra generates the variety of all MV-algebras.
compact and modules
The design was so compact that cooling and servicing the CPU modules proved effectively impossible.
The Roland TB-303 bass synthesizer These producers were aided in their efforts by the availability of affordable, mass-produced electronic music instruments, including synthesizers, compact sequencers, drum machines ( like the Roland TR-909, TR-808 and TR-707, and Latin percussion machine the TR-727 ) and bass modules ( such as the Roland TB-303 ).
Overlays provide a simple means to augment the functionality of a database without requiring that an entirely new backend be written, and allow new functionalities to be added in compact, easily debuggable and maintainable modules.
In the mathematical fields of topology and K-theory, the Serre – Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: " projective modules over commutative rings are like vector bundles on compact spaces ".
When the Leclerc was introduced in the early nineties this was still considered adequate, due to the larger thickness of its modules compared to the armour of other modern western tanks, made possible — for a given weight limit — by the compact design of the tank as a whole.
In mathematics, especially in the area of abstract algebra known as module theory, algebraically compact modules, also called pure-injective modules, are modules that have a certain " nice " property which allows the solution of infinite systems of equations in the module by finitary means.
These algebraically compact modules are analogous to injective modules, where one can extend all module homomorphisms.
All injective modules are algebraically compact, and the analogy between the two is made quite precise by a category embedding.
This gives rise to the intuition that algebraically compact modules are those ( possibly " large ") modules which share the nice properties of " small " modules.
Many algebraically compact modules can be produced using the injective cogenerator Q / Z of abelian groups.
Furthermore, there are pure injective homomorphisms H → H **, natural in H. One can often simplify a problem by first applying the *- functor, since algebraically compact modules are easier to deal with.
When deflated, inflatable modules provide an ' easier to launch ' compact form.
Some SFF computers even include compact components designed for mobile computers, such as notebook optical drives, notebook memory modules, notebook processors, and external AC adapters, rather than the internal power supply units found in full-size desktop computers.
compact and share
* Topological manifolds share the local properties of Euclidean spaces and are therefore also all locally compact.
Philips responded with the digital compact cassette, a system which was backward-compatible with existing analog cassette recordings for playback, but it failed to garner a significant market share and was withdrawn.
Though he signed the compact, Halifax claimed no share in the harsh terms imposed upon the Dutch, and henceforth became a bitter opponent of the policy of subservience to French interests and of the Roman Catholic claims.
After Ellison ordered a new and more compact yacht, he sold his half share in Rising Sun to Geffen in 2010.
The compact disc, invented by Philips and Sony, helped greatly in boosting the company's sales and market share.
The governor opposed ratification of the compact, which appropriated water rights to the Colorado River among seven states, claiming that it gave California an unfair share of " Arizona's birthright ".
In North America, currently the compact segment has a significant share of 16 % of US market.
The sport compact, the Super Bee that share the F-Body platform was also descontinued, so Chrysler of Mexico had to remplace it with a new sport car.
They share the general body plan of a squat lobster, but their bodies are more compact and flattened, an adaptation for living and hiding under rocks.
These new technologies allow individuals to share their stories over the Internet on YouTube, Vimeo, compact discs, podcasts, and other electronic distribution systems.
The task of the Dedra was demanding: it must gather the legacy of Prisma, which relaunched the Lancia brand in the field of compact executive cars, and expand its market share if possible.
He campaigned for Mayor of York in 1988, describing the city's government as a " family compact " and promising to fight for an increased share of Metropolitan Toronto's tax base.
compact and many
Staley Hanover ( Knight Dream-Sweetmite Hanover ) is a little on the small side but a very compact colt and looks like one to stand training and many future battles with colts in his class.
Publisher Richardson has updated the Blue Book `` but it still remains the compact reference book used by so many for those ever-changing telephone numbers, addresses, other residences, club affiliations and marriages ''.
There are also cycling specific multi-tools that combine many of these implements into a single compact device.
During this period, a method of manufacturing many interconnected transistors in a compact space was developed.
In spaces that are compact in this latter sense, it is often possible to patch together information that holds locally — that is, in a neighborhood of each point — into corresponding statements that hold throughout the space, and many theorems are of this character.
In this fashion, one can prove many important theorems in the class of compact spaces, that do not hold in the context of non-compact ones.
Specifically, a topological space is compact if, whenever a collection of open sets covers the space, some subcollection consisting only of finitely many open sets also covers the space.
Slightly more generally, any space with a finite topology ( only finitely many open sets ) is compact ; this includes in particular the trivial topology.
Eventually, the use of horse-mounted cavalry would decline, but carbines continued to be issued and used by many who preferred a lighter, more compact weapon even at the cost of reduced long-range accuracy and power.
These same visible lines, accompanied by the emission lines of trivalent europium and trivalent terbium, and further accompanied by the emission continuum of divalent europium in the blue region, comprise the more discontinuous light emission of the modern trichromatic phosphor systems used in many compact fluorescent lamp and traditional lamps where better color rendition is a goal.
However, there are also many computer magazines or newspapers that provide ratings for freeware and include compact discs or other storage media containing freeware.
Proponents argue that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory and relativity.
In many compact Mishnah printings, a condensed version of his commentary, titled Ikar Tosafot Yom Tov, is featured.
Because it requires no battery or other source of energy, the magneto is a compact and reliable self-contained ignition system, which is why it remains in use in many general aviation applications.
A micro-controller instruction set usually has many instructions intended for bit-wise operations to make control programs more compact.
Specialized folding keyboards, which offer a full-sized keyboard but collapse into a compact size for transport, are available for many models.
While there are many audiophiles who still prefer vinyl records over digital music sources ( primarily compact disc ) for superior sound quality, they represent an enthusiastic minority of listeners.
Downtown Pittsburgh is tight and compact, featuring many skyscrapers, nine of which top.
Essentially a shortened. 222 Remington, the compact. 221 Fireball delivered accuracy exceeding many rifles, out to ranges unheard of for other handguns.
The V6 is one of the most compact engine configurations, shorter than the inline-4 and in many designs narrower than the V8.
William granted some lands to his continental followers from the holding or holdings of one or more specific Englishmen ; at other times, William granted a compact grouping of lands previously held by many different Englishmen to one Norman follower, often to allow for the consolidation of lands around a strategically placed castle.
Dodge had offered panel delivery models for many years since its founding, but their first purpose-built van model arrived for 1964 with the compact A Series.
The XPath syntax comes in two flavours: the abbreviated syntax, is more compact and allows XPaths to be written and read easily using intuitive and, in many cases, familiar characters and constructs.
The compact notation allows many defaults and abbreviations for common cases.
Because of their inefficiency, incandescent light bulbs are gradually being replaced in many applications by other types of electric lights, such as fluorescent lamps, compact fluorescent lamps ( CFL ), cold cathode fluorescent lamps ( CCFL ), high-intensity discharge lamps, and light-emitting diodes ( LEDs ).
4.018 seconds.