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 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 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 1990s, scholars including Stanley Payne, Roger Eatwell, Roger Griffin and Robert O. Paxton have been gathering a rough consensus on the ideology's core tenets.

Since the Hubble sequence is entirely based upon visual morphological type, it may miss certain important characteristics of galaxies such as star formation rate ( in starburst galaxies ) and activity in the core ( in active galaxies ).

Although catalysis of the peptide bond involves the C2 hydroxyl of RNA's P-site ( see Function section below ) adenosine in a protein shuttle mechanism, other steps in protein synthesis ( such as translocation ) are caused by changes in protein conformations. Since their catalytic core is made of RNA, ribosomes are classified as " ribozymes ," and it is thought that they might be remnants of the RNA world.

Since S-waves do not pass through liquids, the liquid core causes a " shadow " on the side of the planet opposite of the earthquake where no direct S-waves are observed.

Since in the past core memory was at a premium, only highly used programs ran 100 % of the time as core resident, most ran as file resident.

Since core memory is monetarily cheap and physically much much larger, greater numbers of programs could be allocated to reside in core.

However, C language programming is much easier to obtain skilled people in, so most if not all new development is done in C. Since z / TPF allows assembler programs to be repackaged into 1 logical file, critical legacy applications can be maintained and actually improve efficiency-the cost of entering one of these programs will now come at the initial enter when the entire program is fetched into core and logical flow through the program is accomplished via simple branch instructions, instead of a dozen or so IBM instructions previously needed to perform what is known as ' core resident enter / back '.

Since the efficiency of a rocket engine is related to the square root of the temperature of the working fluid, the solid core design needs to be constructed of materials that remain strong at as high a temperature as possible.

Since 1991, most of the territory forms the core of Moldova, with smaller parts in Ukraine.

Since the early minicomputers used magnetic core memory, which did not lose its information when power was off, these bootstrap loaders would remain in place unless they were erased, which sometimes happened accidentally when a program bug caused a loop that overwrote all of memory.

Since JSF is primarily component oriented, this is one of the core packages.

Since the beginning, the design and production of military trainers have been Alenia Aermacchi's core business.

Since the core credit card product is essentially identical, the only remaining market differentiators are branding & customer service.

Since the Great Lakes region was the core of Iroquoian-language speakers, scholars have theorized that the Cherokee migrated South from that region.

Since the core S / 360 instruction set remained geared to a 24-bit universe, this third step would require a real break from the status quo ; existing assembly language applications would of course not benefit, and new compilers would be needed before non-assembler applications could be migrated.

Since then, in its campaigns and undercover investigations, it has focused on four core issues — opposition to factory farming, fur farming, animal testing, and animals in entertainment — though it also campaigns against fishing, the killing of animals regarded as pests, the keeping of chained backyard dogs, cock fighting, dog fighting, and bullfighting.

Since 2000, more and more initiatives have been taken at the international and national levels in order to strengthen national health systems as the core components of the global health system.

Since the logistics company had the production plates for key products such as core D & D books, there was no means of printing or shipping core products to generate income or secure short-term financing.

Since core memory was very expensive at the time, several parts of the TX-0 memory were cannibalized for the TX-2 project.

Since 1944 he has also conducted regularly at the San Francisco Opera, where he made his debut with a memorable performance of Verdi's Falstaff.

Since 1954 the Negro's desire for social justice has led to an ironically anarchical rebellion.

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 1953 California has led the nation in enacting guarantees that public business shall be publicly conducted, but not until this year did the lawmakers in Sacramento plug the remaining loopholes in the Brown Act.

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 it was issued in the spring of 1611, the King James Version has been most generally considered the most poetic and beautiful of all translations of the Bible.

Since the Connally amendment has the effect of giving the same right to the other party to a dispute with the United States, it also prevents us from using the court effectively.

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 then nothing has happened to save the life of Pope.

Since accurate base maps are necessary for any planning program, the first step taken by the planning division to implement the long-range state plan has been to prepare two series of base maps -- one at a scale of 1 inch to a mile, and the second a series of 26 sheets at a scale of 1 inch to 2000 feet, covering the entire state.

Since then, there has been a notable increase in the number of stations and also the accumulation of additional data and the development of new techniques for using it, leading to a better understanding of propagation phenomena.

Since the Industrial Revolution, when factories emerged, this classical pattern has been followed.

Since Af are linearly independent functions and the exponential function has no zeros, these R functions Af, form a basis for the space of solutions.

Since C is rational, this correspondence has K coincidences, each of which implies a line of the pencil which meets its image.

Since the course, one member has lost her husband.

Since her bereavement this individual has reported to the writer on numerous occasions about how helpful the class discussions were to her in this adjustment crisis.

Since on the one hand school desegregation has come in Virginia hand-in-glove with pupil assignment, shall we support the plan??

Since this book is concerned only incidentally with railroad rates, it will not attempt to analyze the methods by which the staff of the Interstate Commerce Commission has estimated out-of-pocket costs and apportioned residue costs.

Since that time the telegraph office has shifted in location from the railroad station at the Depot and shops at the Center back to the town clerk's office and drugstore at the Village.

Since 1948 the working headquarters has been the Department of Geology and Mineralogy.

Since the removal force is a function of coating thickness, a differential transformer pickup has been incorporated into the instrument to accurately measure film thickness.

Since the mid 1950s, when urethane foam first made its appearance in the American market, growth has been little short of fantastic.

Since interviewing is the basic therapeutic and diagnostic instrument of modern psychiatry, the recording of interviews for playbacks and study has been a boost of Redstone proportions in new research and training.

Since 1949, the St. Louis club has been below on May 1 just four times.

Since 1927, fewer than a dozen men have made serious runs at Babe Ruth's record and each, in turn, has been thwarted.