* K < nowiki ></ nowiki > X < nowiki ></ nowiki >, the ring of formal power series over the field K. For each nonzero power series P, define f ( P ) as the degree of the smallest power of X occurring in P. In particular, for two nonzero power series P and Q, f ( P )≤ f ( Q ) iff P divides Q.

A polynomial f in one variable X over a ring R is defined as a formal expression of the form

) Furthermore, a ring of formal power series in one variable over a field is a PID since every ideal is of the form.

In a formal wedding, the ring bearer is a special page who carries the rings down the aisle.

The set of all formal power series in X with coefficients in a commutative ring R form another ring that is written R < nowiki ></ nowiki > X < nowiki ></ nowiki >, and called the ring of formal power series in the variable X over R.

This is called the ring of formal power series over R and is denoted by R < nowiki ></ nowiki > X < nowiki ></ nowiki >.

This applies in particular when the base ring R already comes with a topology other than the discrete one, for instance if it is also a ring of formal power series.

Consider the ring of formal power series

This asymmetry disappears if the power series ring in Y is given the product topology where each copy of is given its topology as a ring of formal power series rather than the discrete topology.

A formal Laurent series over a ring R is defined in a similar way to a formal power series, except that we also allow finitely many terms of negative degree ( this is different from the classical Laurent series ), that is series of the form

) The formal Laurent series form the ring of formal Laurent series over R, denoted by R (( X )).

If I is an index set and X < sub > I </ sub > is the set of indeterminates X < sub > i </ sub > for i ∈ I, then a monomial X < sup > α </ sup > is any finite product of elements of X < sub > I </ sub > ( repetitions allowed ); a formal power series in X < sub > I </ sub > with coefficients in a ring R is determined by any mapping from the set of monomials X < sup > α </ sup > to a corresponding coefficient c < sub > α </ sub >, and is denoted.

The set of all such formal power series is denoted R < nowiki ></ nowiki > X < sub > I </ sub >< nowiki ></ nowiki >, and it is given a ring structure by defining

As remarked above, the topology on a repeated formal power series ring like R < nowiki ></ nowiki > X < nowiki ></ nowiki > Y < nowiki ></ nowiki > is usually chosen in such a way that it becomes isomorphic as a topological ring to R < nowiki ></ nowiki > X, Y < nowiki ></ nowiki >.

Most importantly Abdallah gave Abd al-Rahman his ring, the symbol of power, when Abdallah fell ill prior to his death.

The Emperor renounced the right to invest ecclesiastics with ring and crosier, the symbols of their spiritual power, and guaranteed election by the canons of cathedral or abbey and free consecration.

A modern restatement of the theorem in algebraic language is that for a positive integer with prime factorization we have the isomorphism between a ring and the direct product of its prime power parts:

The ring, engraved with the Seal of Solomon, had the power to bind demons from doing harm.

This meaning survived into the early nineteenth century: during a case heard in London in 1824 regarding the legal validity of a marriage ceremony consisting of nothing more than the groom placing a ring on the bride's finger before witnesses, a court official commented that the ceremony " amounted to nothing more than a broomstick marriage, which the parties had it in their power to dissolve at will.

Metallo was soon kidnapped and his power source forcibly removed by Lex Luthor who, after discovering its debilitating effects on Kryptonians, created a ring with a kryptonite gem to keep Superman at bay.

However, Luthor escapes after receiving a power ring fueled by the orange light of avarice and becomes a deputy of the Orange Lanterns.

However, the Agent Orange Larfleeze wants Luthor's ring off, as the alien does not want to share his power with him, resulting in them battling each other for it despite all of the dangers around them.

In the aftermath of the crisis, Luthor, craving the power of the orange light, recovers and operates on several Black Lantern remains in an attempt to find one with a ring still on its finger.

Written by Paul Cornell, the first story entitled " The Black Ring " explores Luthor's more aggressive lust for power in the wake of his exposure to a power ring in the Blackest Night event.

During the Blackest Night crossover, he wore an orange power ring of avarice.

The Emperor renounced the right to invest ecclesiastics with ring and crosier, the symbols of their spiritual power, and guaranteed election by the canons of cathedral or abbey and free consecration.

A magic ring called the " Seal of Solomon " was supposedly given to Solomon and gave him power over demons.

Both wire-service and private teleprinters had bells to signal important incoming messages and could ring 24 / 7 while the power was turned on.

With the Champion limited to his own power, without an omnipotent external power-source, and the She-Hulk having trained to increase both her strength and combat skills in the time between her first trial and the appeal, the Champion was swiftly dealt his first known defeat inside the ring.

Others have special weapons or technology, such as Iron Man's powered armor suits and Green Lantern ’ s power ring.

Stepping out of the ring into their corners, each wrestler is given a ladleful of water, the chikara-mizu (" power water "), with which he rinses out his mouth ; and a paper tissue, the chikara-gami (" power paper "), to dry his lips.

A series of ring roads were constructed ( Isa al Kabeer avenue in the 1930s, Exhibition avenue in the 1960s and Al Fateh highway in the 1980s ), to push back the coastline and extend the city area in belt-like forms.

The fluxmeter element consists of a ring of several thermocouples in series.

The soot is formed through a series of complex chemical reactions, leading from the fuel molecule through molecular growth, until multi-carbon ring compounds are formed.

Also, R < nowiki ></ nowiki > X < nowiki ></ nowiki >, the power series ring is a Noetherian ring.

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.

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.

Just how many sub secrets were being handed over when the ring, watched for six months, was broken remained untold.

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 A is an algebra over its center Z ( A ), or over any subring of its center.

* Any commutative ring R is an algebra over itself, or any subring of R.

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.

The shaft was fitted into the socket of the fore shaft and a bone ring was then placed over the joint to hold the two pieces together, as well as, protecting the wooden shaft from splitting.

The projective plane over any alternative division ring is a Moufang plane.

) folded or arched metal ring attached to a thick wood rim, over which a skin, or most often, a plastic membrane ( or head ) is stretched-it is the bell bronze that gives the banjo a crisp powerful lower register and clear, bell-like treble register-especially in bluegrass music.

The definition works without any changes if instead of vector spaces over a field F, we use modules over a commutative ring R. It also can be easily generalized to n-ary functions, where the proper term is multilinear.

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

Crocheted fabric is begun by placing a slip-knot loop on the hook ( though other methods, such as a magic ring or simple folding over of the yarn may be used ), pulling another loop through the first loop, and repeating this process to create a chain of a suitable length.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

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.