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.

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.

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.

Considered as a ring, however, it has only the trivial automorphism.

The arene ring has an ability to stabilize charges.

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.

An associative R-algebra is an additive abelian group A which has the structure of both a ring and an R-module in such a way that ring multiplication is R-bilinear:

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

Nora leaves her keys and wedding ring and as Torvald breaks down and begins to cry, baffled by what has happened, Nora leaves the house, slamming the door behind herself.

Each fighter has an assigned corner of the ring, where his or her coach, as well as one or more " seconds " may administer to the fighter at the beginning of the fight and between rounds.

When a boxer is knocked down, the other boxer must immediately cease fighting and move to the furthest neutral corner of the ring until the referee has either ruled a knockout or called for the fight to continue.

Bayonne has the longest tradition of bull-fighting in France and there is a ring beyond the walls of Grand Bayonne.

A new ring road has been built from Praia International Airport around the city of Praia.

It has a central core made of older stars that resembles an elliptical galaxy, and an outer ring of young stars that orbits around the core.

This galaxy has also been cited in studies of dark matter, because the stars in the outer ring orbit too quickly for their collective mass.

Though astronomers are not sure what has caused this ring of new stars, some hypothesize that it is from shock waves caused by a bar that is thus far invisible.

If a compression test does give a low figure, and it has been determined it is not due to intake valve closure / camshaft characteristics, then one can differentiate between the cause being valve / seat seal issues and ring seal by squirting engine oil into the spark plug orifice, in a quantity sufficient to disperse across the piston crown and the circumference of the top ring land, and thereby effect the mentioned seal.

If the sheet contains regions where the number of atoms in a ring is different from six, while the total number of atoms remains the same, a topological defect has formed.

Some authors also require the domain of the Euclidean function be the entire ring R ; this can always be accommodated by adding 1 to the values at all nonzero elements, and defining the function to be 0 at the zero element of R, but the result is somewhat awkward in the case of K. The definition is sometimes generalized by allowing the Euclidean function to take its values in any well-ordered set ; this weakening does not affect the most important implications of the Euclidean property.

The property ( EF1 ) can be restated as follows: for any principal ideal I of R with nonzero generator b, all nonzero classes of the quotient ring R / I have a representative r with.

Corresponding to the Bezout property we may, in any commutative ring, consider the collection of elements of the form pa + qb, where p and q range over the ring.

The ring only becomes the property of the woman when marriage occurs.

A ring offered in the form of a Christmas present will likely remain the personal property of the recipient in the event of a breakup.

The fundamental property of used in the proof above is that there cannot be a chain of positive integers where each integer in the chain is strictly less than its predecessor ; in other words, the ring of integers is not " too large " since it cannot sustain such a " large chain ".

The Noetherian property is central in ring theory and in areas that make heavy use of rings, such as algebraic geometry.

In this case the " bad " property is that these elements annihilate all simple left and right modules of the ring.

The Jacobson radical of a ring consists of elements which satisfy a stronger property than being merely a non-unit – in some sense, a member of the Jacobson radical must not " act as a unit " in any module " internal to the ring.

The ring R < nowiki ></ nowiki > X < nowiki ></ nowiki > may be characterized by the following universal property.

In 1861, it was used as a parade ring for the first State Fair held on the Rinearson property, with the Pow-Wow Tree marking the entrance.

Subsequently, a mitigation measure to offset the loss of city green space has drawn criticism for using city property which is technically within city limits, but fall outside the city's leevee ring.

Another illustration of the delicate / robust dichotomy is the fact that being a Dedekind domain is, among Noetherian domains, a local property -- a Noetherian domain is Dedekind iff for every maximal ideal of the localization is a Dedekind ring.

The ring of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one ( to see the last property, observe that for any nonzero ideal I of R, R / I is finite and recall that a finite integral domain is a field ), so by ( DD4 ) R is a Dedekind domain.

It was shown that while rings of algebraic integers do not always have unique factorization into primes ( because they need not be principal ideal domains ), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals ( that is, every ring of algebraic integers is a Dedekind domain ).

However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.

The fourth property can be paraphrased as follows: a ring R is local if and only if there do not exist two coprime proper ( principal ) ( left ) ideals where two ideals I < sub > 1 </ sub >, I < sub > 2 </ sub > are called coprime if R = I < sub > 1 </ sub > + I < sub > 2 </ sub >.

In organic chemistry, aromaticity is a chemical property describing the way in which a conjugated ring of unsaturated bonds, lone pairs, or empty orbitals exhibits a stabilization stronger than would be expected by the stabilization of conjugation alone.

Another example of a presheaf that fails to be a sheaf is the constant presheaf that associates the same fixed set ( or abelian group, or a ring ,...) to each open set: it follows from the gluing property of sheaves that sections on a disjoint union of two open sets is the Cartesian product of the sections over the two open sets.

Since the product of units is a unit and since ring homomorphisms respect products, we may and will assume that S is a submonoid of the multiplicative monoid of R, i. e. 1 is in S and for s and t in S we also have st in S. A subset of R with this property is called a multiplicatively closed set or, shortly, multiplicative set.

** Commutative semigroup, commutative monoid, abelian group, and commutative ring, algebraic structures with the commutative property