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Page "Preadditive category" ¶ 10
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ring and is
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.
Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.
* A field automorphism is a bijective ring homomorphism from a field to itself.
The same definition holds in any unital ring or algebra where a is any invertible element.
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.

ring and endomorphism
* endomorphism ring
* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).
In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring ; every division ring arises in this fashion from some simple module.
In the case when the endomorphism ring, where each endomorphism arises as left multiplication by a fixed ring element.
If a is an idempotent of the endomorphism ring End < sub > R </ sub >( M ), then the endomorphism is an R module involution of M. That is, f is an R homomorphism such that f < sup > 2 </ sup > is the identity endomorphism of M.
Consequently the endomorphism ring of any simple module is a division ring.
For example, the Z-module Q is not simple, but its endomorphism ring is isomorphic to the field Q.
For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.
Focusing on a single object A in a preadditive category, these facts say that the endomorphism hom-set Hom ( A, A ) is a ring, if we define multiplication in the ring to be composition.

ring and .
In the brief moment I had to talk to them before I took my post on the ring of defenses, I indicated I was sickened by the methods men employed to live and trade on the river.
For a blood-chilling ring of terror to the very sound of his name was the tool he needed for the job he'd promised to do.
Opposite every gate was a hitching post or a stone carriage-step, set with a rusty iron ring for tying a horse.
When they were first written, there was evidently no thought of their being published, and those which refer to the writer's love for Mrs. Meynell particularly have the ring of truth.
Both abolition of war and new techniques of production, particularly robot factories, greatly increase the world's wealth, a situation described in the following passage, which has the true utopian ring: `` Everything was so cheap that the necessities of life were free, provided as a public service by the community, as roads, water, street lighting and drainage had once been.
Her mother, now dead, was my good friend and when she came to tell us about her plans and to show off her ring I had a sobering wish to say something meaningful to her, something her mother would wish said.
One means to help the birds occurs to me: Let the chimes that ring over Washington Square twice daily, discontinue any piece of music but one.
The first shrill ring of the telephone brought Harold out into the hall.
Hearing his voice ring raucously up from the road, Kate would await him anxiously and watch perplexed as he walked into the house, cold sober.
The morning hawk, hungry for any eatable, killable, digestible item, kept his eyes on the ring of anchored ships that lay off the shores in the bay, sheltered by the Jersey inlets.
When you and your dog step into the Junior ring, it should be just what the dog wants to do as much as what you want him to do.
Before the Juniors entered the ring the Steward announced that after all Juniors had moved their dogs around the ring and set them up, they could relax with their dogs.
As the Juniors entered the ring, Mr. Spring, the announcer, stated over the public-address system that this was the 28th year that Westminster has held the Finals of the Junior Competition.
After the judge moved all the dogs individually, she selected several from the group and placed them in the center of the ring.
When directions read `` sl a marker on needle '', put a small safety pin, paper clip, or commercial ring marker on needle.
The energy received by the `` pill '' causes the resonant circuit to `` ring '' on after the burst and this `` ringing '' takes place at the resonant frequency of the `` pill ''.
This gave her an opportunity to ring down the curtain with the petulant admonition that we should not presume to lecture her on Abstraction.
( Jelke later served 21 months when he was found guilty of masterminding a ring of high-priced call girls.
Houston police got to know Diane two years ago when the vice squad picked her up for questioning about a call girl ring.
He dialed it and listened to it ring ten times before he hung up.
-- Vital secrets of Britain's first atomic submarine, the Dreadnought, and, by implication, of the entire United States navy's still-building nuclear sub fleet, were stolen by a London-based soviet spy ring, secret service agents testified today.
Just how many sub secrets were being handed over when the ring, watched for six months, was broken remained untold.

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