In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

* Any ring of matrices with coefficients in a commutative ring R forms an R-algebra under matrix addition and multiplication.

In addition to the 121 Chadian soldiers in the Multinational Force in the Central African Republic ( FOMUC ), there are still 150 soldiers from Chad in the C. A. R.

In addition to being an upright bassist, he wrote dozens of R & B hits and worked as a producer.

In addition, there was an insistence on women being allowed to participate in the Order in " perfect equality " with men, in contrast to the S. R. I. A.

In addition, the most recent data show that Italy's spending in R & D in 2011 was equal to 1. 1 % of GDP ( 12th in the world by expenditures ), below the European average of 1. 7 % and the Lisbon Strategy target of devoting 3 % of GDP to research and development activities.

In addition, for any rate R > C, it is impossible to transmit with arbitrarily small block error.

* Euclidean space R < sup > n </ sup > with ordinary vector addition as the group operation becomes an n-dimensional noncompact abelian Lie group.

In addition, both large and small companies work in R & D to explore MEMS technology.

Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements.

In addition, R. J. Reynolds ( the tobacco conglomerate ) took over as the major sponsor of NASCAR racing ( changing the name to the " Winston Cup ") and they made a significantly larger financial contribution than previous sponsors.

The real numbers R, together with addition as operation and its usual topology, form a topological group.

More generally, Euclidean n-space R < sup > n </ sup > with addition and standard topology is a topological group.

A ring is a set R equipped with two binary operations +: R × R → R and ·: R × R → R ( where × denotes the Cartesian product ), called addition and multiplication.

* ( R, +) is required to be an abelian group under addition:

In addition to providing the illustrations for her own Moomin books, Jansson also illustrated Swedish translations of classics such as J. R. R. Tolkien's The Hobbit and Lewis Carroll's The Hunting of the Snark and Alice's Adventures in Wonderland ( some used later in Finnish translations as well ).

The transfer principle states that true first order statements about R are also valid in * R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals ; since R is a real closed field, so is * R. Since for all integers n, one also has for all hyperintegers H. The transfer principle for ultrapowers is a consequence of Łoś ' theorem of 1955.

Montero had set up a strong position, using every bale and box we had in addition to barricades of logs and brush.

They'd peddled the soap virtually alone, and without much success, until about a year ago, when -- with the addition of `` SX-21 '' to their secret formula and the inauguration of a high-powered advertising campaign -- sales had soared practically into orbit.

Present at the scene -- in addition to the dead man, who was indeed Louis Thor -- had been Thor's partner Bill Blake, and Antony Rose, an advertising agency executive who handled the zing account.

In addition to the authentication and acknowledgment procedures which precede and follow the sending of the go messages, again in special codes, each message also contains an `` internal authenticator '', another specific signal to convince the recipient that he is getting the real thing.

In Frankfurt, too, he directed the Museum and Opera House concerts which, in addition to the standard repertoire, featured novelties like Erdmann's Piano Concerto and Mahler's Sixth Symphony.

In addition, our way of dealing directly with natural phenomena has also changed.

In addition, they have been converted to Zen Buddhism, with its glorification of all that is `` natural '' and mysteriously alive, the sense that everything in the world is flowing.

The networks for military communications are one of the best examples of networks which not only must be changed with the changes in objectives but also must be changed with the addition of new machines of war.

Besides showing no inclination, apparently, to absent himself from his native region even for short periods, and in addition writing a shelf of books set in the region, he has handled in those books an astonishingly complete list of matters which have been important in the South during the past hundred years.

In addition, he believed in the `` dramatic unity and separateness of the period from 1702-14, lying between the Stuart and Hanoverian eras with a special ethos of its own ''.

In addition, his definition of a tragedy invites our attention, because a serious and important action may very well be one that tests the moral fiber of the author or of the characters.

In addition, the 1952 study collected comparable data from 4,585 students at ten other colleges and universities scattered across the country: Dartmouth, Harvard, Yale, Wesleyan, North Carolina, Fisk, Texas, University of California at Los Angeles, Wayne, and Michigan.

There was much sickness in the corps, and the men were, in addition, without the clothing, shoes, and blankets needed for the winter weather.

In addition, there are many areas of the human situation besides the impact of science and technology which are examined, for science-fiction dystopias often extrapolate political, social, economic tendencies only indirectly related to science and technology.

In San Francisco he has worked with Brew Moore, Charlie Mingus, and other `` swinging '' musicians of secure reputation, thus placing himself within established jazz traditions, in addition to being a part of the San Francisco `` School ''.

In addition to his experiments in reading poetry to jazz, Patchen is beginning to use the figure of the modern jazz musician as a myth hero in the same way he used the figure of the private detective a decade ago.

In addition Rep. Frelinghuysen's brother Harry was on the Korean desk of the State Department in World War 2.

In addition, he said, he has answered more than 400 messages of congratulations which led him to the comment that he himself had decided he wouldn't send another congratulatory message for the rest of his life.

In addition, the motor has the seal of approval of the Underwriters Laboratories, which means it is safe.

In addition, it is small enough to get into crevices, jacket and crown margins, malposed anteriors, and the back teeth.

Warren and White each carried, in addition to their own weapons and ammo, a box of ammo for the ROK's machine gun.

And this in addition to his usual fear of being among people of high society, his fear of making some inane or inappropriate remark.

In addition, the SBA uses such criteria as number of employees and dollar volume of the business.

Thus, there has come into being a situation in which the state must raise all of its own revenues and, in addition, must give assistance to its local governments.

In addition to the regular schedule, advertisements were run for maximum impact in special editions of the New York Times, Boston Herald, American Banker, Electronic News and, for local promotion, the Providence Sunday Journal.