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 a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

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 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 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 ).

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.

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

Since the magnetic field is proportional to the product NI, the number of turns in the windings N and the current I can be chosen to minimize heat losses, as long as their product is constant.

Since the power dissipation, P = I < sup > 2 </ sup > R, increases with the square of the current but only increases approximately linearly with the number of windings, the power lost in the windings can be minimized by reducing I and increasing the number of turns N proportionally.

Since the short-circuited rotor windings have small resistance, a small slip induces a large current in the rotor and produces large torque.

Since the number of windings that can be supported in such armature is small, the output voltage level is correspondingly small.

Since the volts-per-turn is the same in both windings, each develops a voltage in proportion to its number of turns.

Since the homosexuals widely use marijuana, they do not have to be initiated.

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 Rhode Island at that time did not have such sanction, his opinion was not popular.

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 1957, more and more trans-Atlantic passengers have been crossing by air.

Since two of these could be trustees of the college, actually it would be necessary to have the consent of only one elected official to impose a levy of millions of dollars of tax revenue.

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

`` Since the war I have seen the way the Germans have behaved and I think, Holy Mother, we have behaved like this for hundreds of years.

Since that time the demands of the citizens for new and expanded services have placed financial burdens on the state which could not have been foreseen in earlier years.

Since faculty see themselves as self-employed professionals rather than as employees, enthusiasm in a common enterprise is proportionate to the sense of ownership they have in it by virtue of sharing in the decisions that govern its course.

Since the concept of high fidelity became important some dozen years ago, the claims of technical improvements have multiplied tenfold.

Since SAAMI's conception in 1926, and more specifically since the adoption of the Shooting Development Program in 1954, millions of dollars and promotional man-hours have gone into the development of more places to shoot for more youths and adults.

Says NRLDA's James Wright: `` Since 1958 carriers that move material from the yard to the job site have undergone more radical changes than any of the dealer's other equipment ''.

Since practical washing procedures are both lengthy and expensive, a number of laboratory tests have been developed for the numerical evaluation of detergents.

Since there is a continual loss of micrometeoritic material in space because of the radiation effects, there must be a continual replenishment: otherwise, micrometeorites would have disappeared from interplanetary space.

Since Af and P divides Af for Af, we have Af.

A final class of exceptional lines is identifiable from the following considerations: Since no two generators of Af can intersect, it follows that their image curves can have no free intersections.

Since they are all either rented or borrowed, the requested dates for their use have to be far in advance.

Since the writer had not noticed this characteristic in married students scattered throughout the various sections previous to this experiment, nor, as a matter of fact, in those who were continuing in `` single sections '', he can only conclude that there must have been something `` contagious '' within the specific group which caused this to occur.

Since the details of the elections were settled the change of government had no direct effect on the technical aspects of the elections, and may have been more important as an indication of royal displeasure with the U.N.F.P.

Since 1917 some thirteen supplementary and related acts have extended this Federal program.

Since this type of item was not in the statute when section 381 was enacted in 1954, one cannot say with certainty what effect the enactment of that section should have.