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.

Since all maximal paths have the same number of black nodes, by property 5, this shows that no path is more than twice as long as any other path.

Since a one sided maximal ideal A is not necessarily two-sided, the quotient R / A is not necessarily a ring, but it is a simple module over R. If R has a unique maximal right ideal, then R is known as a local ring, and the maximal right ideal is also the unique maximal left and unique maximal two-sided ideal of the ring, and is in fact the Jacobson radical J ( R ).

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F < sub > 2 </ sub >, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Since f ( X ) is irreducible, ( f ( X )) is a maximal ideal and hence is, in fact, a field.

Since as shown above, the classical definition is essentially the modern definition with only maximal ideals considered, this shows that the interpretation of the modern definition as " zero sets of functions " agrees with the classical definition where they both make sense.

Since it is a collection of trees, and trees are contractible, consider the space where the equivalence relation is generated by if they are contained in a common tree in the maximal forest F. The quotient map is a homotopy equivalence.

* Since the conjugate of a maximal torus is a maximal torus, every element of G lies in some maximal torus.

) This follows from the fact that R has a maximal left ideal M, and the fact that the quotient module R / M is a simple left R-module, and that its annihilator is a proper two-sided ideal in R. Since R is a simple ring, this annihilator is

Since the velocity is maximal when all the enzyme is bound as the enzyme-substrate complex,.

Since the compact operators is a ( in fact, the only ) maximal norm-closed ideal in B ( H ), the Calkin algebra is simple.

Since the number of edges in a maximal spanning forest of a graph is equal to the number of vertices minus the number of components, the formula above for the cyclomatic number follows.

Since power is equal to force times velocity, the muscle generates no power at either isometric force ( due to zero velocity ) or maximal velocity ( due to zero force ).

Since the technostructure is composed of an hierarchical system of influential employees inside the enterprise, its primary goal is not to maximize their profits but rather survival, continuous growth and maximal size.

Since the alcohol in wine is produced by fermentation of the sugar, the Oechsle scale is used to predict the maximal possible alcohol content of the finished wine.

Since that time there have been many portraits of Confucius as the ideal philosopher.

Since very few actual implementations of heat engines exactly match their underlying thermodynamic cycles, one could say that a thermodynamic cycle is an ideal case of a mechanical engine.

Since fluorescence emission differs in wavelength ( color ) from the excitation light, an ideal fluorescent image shows only the structure of interest that was labeled with the fluorescent dye.

; Compact size: Since p-code is based on an ideal virtual machine, a p-code program is often much smaller than the same program translated to machine code.

Since the same magnetic flux passes through both the primary and secondary coils in an ideal transformer, the instantaneous voltage across the primary winding equals

Since the ideal core has been assumed to have near-zero reluctance, the magnetizing current is negligible, although still required, to create the magnetic field.

Since the ideal windings have no impedance, they have no associated voltage drop, and so the voltages V < sub > P </ sub > and V < sub > S </ sub > measured at the terminals of the transformer, are equal to the corresponding EMFs.

Since the core flux is proportional to the applied voltage, the iron loss can be represented by a resistance R < sub > C </ sub > in parallel with the ideal transformer.

Since a mass driver could use any type of mass for reaction mass to move the spacecraft, a mass driver or some variation seems ideal for deep-space vehicles that scavenge reaction mass from found resources.

Since a Van de Graaff generator can supply the same small current at almost any level of electrical potential, it is an example of a nearly ideal current source.

Since the topology on R < nowiki ></ nowiki > X < nowiki ></ nowiki > is the ( X )- adic topology and R < nowiki ></ nowiki > X < nowiki ></ nowiki > is complete, we can in particular apply power series to other power series, provided that the arguments don't have constant coefficients ( so that they belong to the ideal ( X )): f ( 0 ), f ( X < sup > 2 </ sup >− X ) and f ( ( 1 − X )< sup >− 1 </ sup > − 1 ) are all well defined for any formal power series f ∈ R < nowiki ></ nowiki > X < nowiki ></ nowiki >.

Since Bheeshma thought the easterners were skilled in fighting from the backs of elephants and in fighting with bare arms, he deemed Bhima to be the most ideal person to wage wars in that region.

Since tall white pines were ideal for use as masts for ships, King George claimed it.

Since some ocarinas are fully chromatic and can be played in professional musical situations, including classical and folk, sheet music is an ideal notation for ocarinas.

Since the Otto cycle is an isentropic process the isentropic equations of ideal gases and the constant pressure / volume relations can be used to yield Equations 3 & 4.

Since antifeedants are nontoxic, they would be ideal as insecticides in agriculture.

Since the castle is situated on a rocky outcrop above the River Mulde, the Germans believed it to be an ideal site for a high security prison.

Since the Supreme Court's decision of that year this is more doubtful ; ;

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

Since it is not far from Viareggio, he will visit Puccini's house, as he never fails to do, to pay his respects to the memory of the composer of La Boheme, which he considers one of Puccini's masterpieces.

Since attack serves to stimulate interest in broadcasts, I added to my opening statement a sentence in which I claimed that German youth seemed to lack the enthusiasm which is a necessary ingredient of anger, and might be classified as uninterested and bored rather than angry.

Since the air is a continuum, the network of communication remains intact regardless of the positions or motions of the points ( the people ) in the net.

Since the hazards of poor communication are so great, p can be justified as a habitable site only on the basis of unusual productivity such as is made available by a waterfall for milling purposes, a mine, or a sugar maple camp.

Since the difficulty of drawing the net is great, we will merely discuss it.

Since a civilizational crisis involves also a crisis in private interests and in the ruling class, reaction is normally found among those who feel themselves to be among the ruling class.

Since civilizational change is the most difficult to perceive and analyze, it seldom is given adequate attention.

Since the slogans have little application to reality and are sanctimonious to boot, the applause is faint even in areas of the world where we should expect to find the greatest affection for free government.

Since more is known about Quiney than about any other acquaintance of Shakespeare in Stratford, his career may be followed to its sudden end in 1602.

Since the great flood of these dystopias has appeared only in the last twelve years, it seems fairly reasonable to assume that the chief impetus was the 1949 publication of Nineteen Eighty-Four, an assumption which is supported by the frequent echoes of such details as Room 101, along with education by conditioning from Brave New World, a book to which science-fiction writers may well have returned with new interest after reading the more powerful Orwell dystopia.

Since Laos is of no more purely military value to Moscow itself than it is to Washington, this approach might be expected to head off Mr. Khrushchev for the moment.

Since little is known about autism, and almost nothing has been written for the layman, we'd like to share one experienced mother's comments.

Since the work is done by hand, the only limitation, it is said, `` is that of human conception ''.

Since the apparatus is new, it requires experimentation and changes in technique.

Since the obvious is not always true, the Republican National Committee wisely analyzed its defeat of last autumn and finds that it occurred, as suspected, in the larger cities.

Since he has just shown who is top dog, he may not be ready to receive this highest honor in the gift of the Soviet people.

Since appeals to morality, to humanity, and to sanity have had such small effect, perhaps our last recourse is the deterrent example.

Since the goal of our international planners is a World Government, this Atlantic Community would mark a giant step in that direction for, once American economic autonomy is absorbed, a larger grouping is a question of time.