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 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.

Since every map from a discrete space is continuous, the topological homomorphisms between discrete groups are exactly the group homomorphisms between the underlying groups.

Since the circle group is compact, any representation ρ: T → GL ( 1, C ) ≅ C < sup >×</ sup >, must take values in U ( 1 ) ≅ T. Therefore, the irreducible representations of the circle group are just the homomorphisms from the circle group to itself.

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 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 we can neither undo nor redo the past, we are limited to the events of today and tomorrow.

Since the recognition of Israel as a nation state, claims are made in many cases which identify the claimant as a member of the new body politic.

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 conventional methods are insensitive at the low frequencies of these molecular transitions, the paramagnetic resonance method is being used instead.

Since broadcast frequencies are very limited in number, these objectives are to some extent inconsistent in that not all of them can be fully realized, and to the extent that each is realized, there is a corresponding reduction of the possibilities for fullest achievement of the others.

Since the provision of skywave service requires adequate freedom from interference, only class 1, stations are capable of rendering skywave service.

Since the sides are also covered up to the spray rails, they are also rough-sanded in that area.

Since fly cutters are one sided and not balanced, they should be used at the slowest speed available, and fed very slowly to avoid binding.

Since there are almost no schools in the country offering graduate work in interior design this rule cannot at present be observed.

Since charges are relatively the same, reserving a car before you leave for Europe will assure you of having one on tap when you want it.

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

Since the circulating thyroid hormones are the amino acids thyroxine and tri-iodothyronine ( cf. Section C ), it is clear that some mechanism must exist in the thyroid gland for their release from proteins before secretion.

Since emotional reactions in the higher vertebrates depend on individual experience and are aroused in man, in addition, by complex symbols, one would expect that the hypothalamus could be excited from the cortex.

Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).

Since D and D' are both diagonalizable and they commute, they are simultaneously diagonalizable, and Af is diagonalizable.

Since N and N' are both nilpotent and they commute, the operator Af is nilpotent ; ;

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 the choice is by lot each week, the outcomes of different trials are independent.

Since two curves of symbol Af on Q intersect in Af points, it follows that there are Af lines of Af which are tangent to Aj.

Since any group is a one-object category, a special case of the question about what the morphisms preserve is this: how to consider any group G as a symmetry group?

Since the morphisms are generated by those that ' skip ' or ' add ' a single element, the detailed relations written out above underlie the topological applications.

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 I have already discussed his moral position, that discussion is incorporated by reference into the following pages, which will focus on the empirical and analytic side of Fromm's treatment.

Since the change to better nutrition, he feels he can report on improvements in health, though he considers the following statements observations and not scientific proof.

Since the Catholic Church expresses such desire that the Sacred Scriptures be read, the following taken from the Holy Bible ( New Catholic Edition ) will prove a means of grace and a source of great spiritual blessing.

Since the administrative reforms in Bavaria in the 1970s, the town also encompasses the following Ortsteile:

Since we require the process to be adiabatic, the following equation needs to be true

Since its video release it has acquired a cult following, along with the other two films in the trilogy.

Since the following two hundred years saw the rise of the Classical Greek civilization, which has been enduringly influential in western society, the Battle of Marathon is often seen as a pivotal moment in European history.

Since the area has been referred to as the Balkans, notable conflicts have included the following

Since 1964, following clashes between the two communities, the Turkish Cypriot seats in the House remained vacant, while the Greek Cypriot Communal Chamber was abolished.

Since the 1980s, following the popularization of computer technology, it has become fashionable in marketing for names of products and companies, and for 1990s online video games where players use pseudonyms ( when spaces were not allowed ).

Since the corona has been photographated at high resolution in the X-rays by the satellite Skylab in 1973, and then later by Yohkoh and the other following space instruments, it has been seen that the structure of the corona is very various and complex: different zones have been immediately classified on the coronal disc

Since the Iraq War, evangelicalism has begun to make inroads in Iraq, capitalizing on a new openness to Western missionaries following the US-led invasion of the country, although the vast majority of converts are from other sects of Christianity present in Iraq .< ref >

Since Arafat's death, he is formally head of Fatah's political bureau and chairman, but his actual political following within Fatah appears limited.

Since July 2007, following the 2006 Palestinian legislative election and the following Battle of Gaza, Hamas has functioned as the de-facto ruler in the Gaza Strip, forming an alternative Hamas Government in Gaza.

Since then, and even in recent years, public spokesmen in Haiti as well as international academics and statesmen have denounced this event as the payment of an illegitimate debt, in several cases calling on the French government to repay it ( the French government has never been willing to repay it, though there was a hoax following the 2010 Haiti Earthquake involving a fake website purporting to offer reparation payment on behalf of the French Government ).

Since 1990, Hezbollah has been accused of the following attacks and attempted attacks.

Since cast iron was becoming cheaper and more plentiful, it also became a major structural material following the building of the innovative Iron Bridge in 1778 by Abraham Darby III.

Since the 1950s, following the lead of D. W. Jefferson, there are those who argue that, whatever its legacy of influence may be, Tristram Shandy in its original context actually represents a resurgence of a much older, Renaissance tradition of " Learned Wit " – owing a debt to such influences as the Scriblerian approach.

Since 1980, linguists studying children, such as Melissa Bowerman, and psychologists following Jean Piaget, like Elizabeth Bates and Jean Mandler, came to suspect that there may indeed be many learning processes involved in the acquisition process, and that ignoring the role of learning may have been a mistake.

Since July 2012, Odyssey has been back in full, nominal operation mode following three weeks of ' safe ' mode on remote maintenance.

Since 2003, following the death in custody of a prisoner, Naseem, the Maldives experienced several anti-government demonstrations calling for political reforms, more freedoms and, and end to torture and oppression.

Since its economy was highly dependent on the rest of the former Soviet Union for energy and raw materials, the breakdown in trade following the breakup of the Soviet Union had a serious effect, exacerbated at times by drought and civil conflict.