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* Every module over a field or skew field is projective ( even free ).
Some Related Sentences
Every and module
Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.
Every library module has at least two source files: a definitions file specifying the library's interface plus one or more program files specifying the implementation of the procedures in the interface.
Every simple R-module is isomorphic to a quotient R / m where m is a maximal right ideal of R. By the above paragraph, any quotient R / m is a simple module.
Every vector space is free, and the free vector space on a set is a special case of a free module on a set.
This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of ( finitely or infinitely many ) copies of this module K < sup > n </ sup >.
Every finite-length module M has a composition series, and the length of every such composition series is equal to the length of M.
Every ring which is semisimple as a module over itself has zero Jacobson radical, but not every ring with zero Jacobson radical is semisimple as a module over itself.
Every superfield, i. e. a field that depends on all coordinates of the superspace ( or in other words, an element of a module of the algebra of functions over superspace ), may be expanded with respect to the new fermionic coordinates.
Every and over
The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.
Every six months the presidency rotates between the states, in an order predefined by the Council's members, allowing each state to preside over the body.
* Every June, Ellensburg hosts Dachshunds on Parade, an event that draws Dachshund dog owners from all over the Northwest.
Every year sidecar racers travel from all over the UK to Knockhill in Scotland to race in the prestigious ' Jock Taylor Memorial Trophy ' race.
Every major commercial transport and almost every military jet built in the United States over the last 40 years has been tested in this facility.
Every car uses either a 5. 0 L Ford " Boss 302 " SVO or a 5. 0 L Chevrolet small block race-engine ( depending on the make )-capable of producing between 460 and 485 kW ( 620 — 650 bhp ) of power, but generally quoted as a little over 450 kW ( 600 bhp ) in race trim.
Every character in the novel — both men and women — knows their place, and the traditional stereotypes of gender roles are repeated over and over again.
Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet.
Every Calan Mai, the two would fight over Creiddylad, until a battle on Judgement Day, in which the victor would keep her forever.
Controversy ensued over the use of the phrase " Every seat lost to the government is a seat sold to the Boers " as the Unionists waged a personalised campaign against Liberal critics of the war – some posters even portrayed Liberal MPs praising President Kruger and helping him to haul down the Union Jack.
Every year, the post office also creates a special Christmas hand-cancellation pictorial postmark for use during December, which also attracts mail from all over the world.
A quotation attributed to Kolmogorov is into English: " Every mathematician believes that he is ahead over all others.
Every August the Mountain Mama Road Bike Challenge brings cyclists from all over the country to challenge themselves on the steep roads of Highland County.
The argument over the quality of Fitzgerald ’ s translation of the Rubaiyat has, according to Dougan, diverted attention from a fuller understanding of the deeply esoteric message contained in Omar ’ s actual material – " Every line of the Rubaiyat has more meaning than almost anything you could read in Sufi literature ".
Every and field
Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.
Every time an MTA receives an email message, it adds a < tt > Received </ tt > trace header field to the top of the header of the message, thereby building a sequential record of MTAs handling the message.
Every ordered field is a formally real field, i. e., 0 cannot be written as a sum of nonzero squares.
Every instruction consists of a 9-bit opcode, a 4-bit register code, and a 23-bit effective address field, which consists in turn of a 1-bit indirect bit, a 4-bit register code, and an 18-bit offset.
Every field theory of particle physics is based on certain symmetries of nature whose existence is deduced from observations.
Every planetary body ( including the Earth ) is surrounded by its own gravitational field, which exerts an attractive force on all objects.
Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.
Every year, the British Political Studies Association awards the Walter Bagehot Prize for the best dissertation in the field of government and public administration.
Every polynomial in can be factorized into polynomials that are irreducible over F. This factorization is unique up to permutation of the factors and the multiplication of the factors by nonzero constants from F ( because the ring of polynomials over a field is a unique factorization domain whose units are the nonzero constant polynomials ).
Every quadratic form q in n variables over a field of characteristic not equal to 2 is equivalent to a diagonal form
Every kind of material has unique magnetic properties, even those that we do not think of as being “ magnetic .” Different materials below the ground can cause local disturbances in the Earth ’ s magnetic field that are detectable with sensitive magnetometers.