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It is a fact that the ring is a principal ideal ring ; that is, for any ideal I in, there exists an integer n in I such that every element of I is a multiple of n. Conversely, the set of all multiples of an arbitrary integer n is necessarily an ideal, and is usually denoted by ( n ).
Some Related Sentences
is and fact
In fact it has caused us to give serious thought to moving our residence south, because it is not easy for the most objective Southerner to sit calmly by when his host is telling a roomful of people that the only way to deal with Southerners who oppose integration is to send in troops and shoot the bastards down.
While the pattern is uneven, some having gained more than others, nationalism has in fact served the Western peoples well.
In point of fact, this is a beige box with a bright red door, about one and a half feet square and hung from the wall about six feet from the door to Wisman's right.
The fact is that the Southern Confederacy differed from the earlier one almost as much as the Federal Constitution did.
To my knowledge, Lincoln remains the only Head of State and Commander-in-Chief who, while fighting a fearful war whose issue was in doubt, proved man enough to say this publicly -- to give his foe the benefit of the fact that in all human truth there is some error, and in all our error, some truth.
In fact the accumulation of the hardware of destruction is day by day increasing our fear of each other.
Operating as a one man police force in fact if not in name, he is at once more independent and more dedicated than the police themselves.
`` I may possibly be a greater risk than is the normal person of my age '', the President had said on February 29th of the election year, ignoring the fact that no one of his age had ever lived out another term.
In the incessant struggle with recalcitrant political fact he learns to focus the essence of a problem in the significant detail, and to articulate the distinctions which clarify the detail as significant, with what is sometimes astounding rapidity.
we accord it its place there, and in Lawrence's treatment we are given the innocent fantasy of a child, in fact, the form in which oedipal love is expressed in childhood.
There is probably some significance in the fact that two of the best incest stories I have encountered in recent years are burlesques of the incest myth.
How much they esteemed him is shown by the fact that their underground committee selected him as one of the few who would be helped to escape.
That is not to deny that he has been aware of traditions, of course, that he is steeped in them, in fact, or that he has dealt with them, in his books.
There is evidence to suggest, in fact, that many authors of the humorous sketches were prompted to write them -- or to make them as indelicate as they are -- by way of protesting against the artificial refinements which had come to dominate the polite letters of the South.
It seems quite obvious that all the really difficult tasks of human beings arise from the fact that man is not one, but many.
If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.
Perhaps the mere fact that by plucking on the nerves nature can awaken in the most ordinary of us, temporarily anyway, the sleeping poet, and in poets can discover their immortality, is the most remarkable of all the remarkable phenomena to which we can attest??
is and ring
Our new large-package ring twister for glass fiber yarns is performing well in our customers' mills.
If you walk into the ring because it is fun to show your dog, he will feel it and give you a good performance!!
`` The reason you are in the ring today is to show your ability to present to any judge the most attractive picture of your dog that the skillful use of your aids can produce.
This is the tale of one John Enright, an American who has accidentally killed a man in the prize ring and is now trying to forget about it in a quiet place where he may become a quiet man.
`` He has married me with a ring of bright water '', begins the Kathleen Raine poem from which Maxwell takes his title, and it is this mystic bond between the human and natural world that the author conveys.
Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.
The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F are those of degree one.
The ceremony of such a blessing is similar in some aspects to the consecration of a bishop, with the new abbot being presented with the mitre, the ring, and the crosier as symbols of office and receiving the laying on of hands and blessing from the celebrant.
The sum, difference, product and quotient of two algebraic numbers is again algebraic ( this fact can be demonstrated using the resultant ), and the algebraic numbers therefore form a field, sometimes denoted by A ( which may also denote the adele ring ) or < span style =" text-decoration: overline ;"> Q </ span >.
If K is a number field, its ring of integers is the subring of algebraic integers in K, and is frequently denoted as O < sub > K </ sub >.
In the context of abstract algebra, for example, a mathematical object is an algebraic structure such as a group, ring, or vector space.
The configuration of six carbon atoms in aromatic compounds is known as a benzene ring, after the simplest possible such hydrocarbon, benzene.
The structure is also illustrated as a circle around the inside of the ring to show six electrons floating around in delocalized molecular orbitals the size of the ring itself.
In aromatic substitution one substituent on the arene ring, usually hydrogen, is replaced by another substituent.
When there is more than one substituent present on the ring, their spatial relationship becomes important for which the arene substitution patterns ortho, meta, and para are devised.
This is seen in, for example, phenol ( C < sub > 6 </ sub > H < sub > 5 </ sub >- OH ), which is acidic at the hydroxyl ( OH ), since a charge on this oxygen ( alkoxide-O < sup >–</ sup >) is partially delocalized into the benzene ring.
is and principal
The principal defender of this view of primary experience as `` causal efficacy '' is Alfred North Whitehead.
but his principal theme is that the intrigues of the Tories, `` our Popish or Jacobite Party '', pose an immediate threat to Church and State.
This is the principal point made in this final section of Englishman No. 57, and it caps Steele's efforts in his other writing of these months to counteract the notion of the Tories as a `` Church Party '' supported by the body of the clergy.
But when the situation was so complicated that even Nogaret, one of the principal actors in the drama, could misinterpret the pope's motives, it is possible that Othon and his companions, equally baffled, attributed their difficulties to a more immediate cause.
A primary function is the operation of a Government Bid Center, which receives bids daily from the Federal Government's principal purchasing agencies.
Subject to the limitations hereinafter provided, the Secretary of the Treasury is authorized and directed to pay, as prescribed by Section 8 of this Title, an amount not exceeding the principal of each award, plus accrued interests on such awards as bear interest, certified pursuant to Section 5 of this Title, in accordance with the award.
If your principal place of abode for the tax year is outside the United States ( including Alaska and Hawaii ), Puerto Rico, or the Virgin Islands and you have no legal residence or principal place of business in any Internal Revenue district in the United States, you should file your return with the Office of International Operations, Internal Revenue Service, Washington 25, D.C..
But in such an important question, we would be satisfied if the judgment were that the principal objection to the identity of forces which produce electricity and magnetism were only a difficulty, and not a thing which is contrary to it.
There are three principal feed bunk types for dairy and beef cattle: ( 1 ) Fence-line bunks -- cattle eat from one side while feed is put in from the opposite side of the fence by self-unloading wagons ; ;
This is a pilot operation sponsored by a new entity chartered in Delaware as the Tri-State Pipeline Corporation, with principal offices in New York State.
Emotional maturity is the result of many factors, the principal ones being the experiences of the first few years of the child's life.
Perhaps one way to sharpen our sense of the modernity of Utopian communism is to contrast it with the principal earlier types of communistic theory.
-- One of the principal aims of anionic polymerization techniques is the synthesis of polymers of extremely narrow molecular weight distribution.
If it is owned, taxes must be paid, and if the place is not free of mortgage, there will be interest and payments on the principal to take care of.
The role of an earthquake in starting the destruction of whole cities is tremendously frightening, but fire may actually be the principal agent in a particular disaster.
In general, friendly contact with a member followed by contact with a clergyman will account for a major share of recruitment by the churches, making it quite evident that the extension of economic integration through co-optation is the principal form of mission in the contemporary church ; ;
The new school superintendent is Harry Davis, a veteran agriculture teacher, who defeated Felix Bush, a school principal and chairman of the Miller County Democratic Executive Committee.
The principal of the school announced that -- despite the help of private tutors in Hollywood and Philadelphia -- Fabian is a 10-o'clock scholar in English and mathematics.
While it must be said that these same Protestants have built some new churches during this period, and that religious population shifts have emptied churches, a principal reason for this phenomenon of redundancy is that fewer Protestants are going to church.
However, my principal objection in this sort of novel is to the hackneyed treatment of race-drivers, pilots, submariners, atomic researchers, and all the machine-masters of our age as brooding mystics or hysterical fatalists.