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By the Chinese Remainder Theorem, this ring factors into the direct product of rings of integers mod p. Now each of these factors is a field, so it is clear that the only idempotents will be 0 and 1.
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By and Chinese
By the mid-to-late 13th century, the Chinese had adopted the dogma of Neo-Confucian philosophy formulated by Zhu Xi.
By 1949, the CPC had established control over most of the country ( see Chinese Civil War ).
By the time of the Tang Dynasty five-hundred years after Buddhisms arrival into China, it had transformed into a thoroughly Chinese religious philosophy dominated by the school of Zen Buddhism.
By this time large Chinese infantry-based armies of 100, 000 to 200, 000 troops were now buttressed with several hundred thousand mounted cavalry in support or as an effective striking force.
By adopting the name Chung-cheng (" central uprightness "), he was choosing a name very similar to the name of Sun Yat-sen, who was ( and still is ) known among Chinese as Zhongshan ( 中山 — meaning " central mountain "), thus establishing a link between the two.
By 1880, almost a thousand Chinese people, " one for every five Caucasians ," lived in Carson City.
Taylor & Taylor ( 1995: 124 ) concluded, " By and large, Pinyin represents the Chinese sounds better than the Wade-Giles system, and does so with fewer extra marks.
By 1906, the total number of Chinese coolies had swelled to 50, 000, almost entirely recruited and shipped by CEMC.
By the turn of the 19th century, the company had become the largest of the ‘’ hongs ’’ or foreign trading conglomerates with offices in all the important Chinese cities as well as Yokohama, Japan.
By the mid-1950s the Chinese army had entered the north-east portion of Ladakh.
By 1921, small groups of Marxists existed in six Chinese cities: Shanghai, Peking, Changsha, Wuhan, Canton and Tsinan, with a further group having been founded by Chinese students in Paris.
By placing the prisoners under conditions of physical and social deprivation and disruption, and then by offering them more comfortable situations such as better sleeping quarters, better food, warmer clothes or blankets, the Chinese did succeed in getting some of the prisoners to make anti-American statements.
By the time of Mozi, Chinese rulers and the wealthier citizens already had the practice of extravagant burial rituals.
By contrast, neighboring Chinese leader Mao Zedong was mostly ignored and Kim Il-sung rejected most of his policies such as the Hundred Flowers Campaign and ( later ) the Cultural Revolution.
By the end of August 2006, statistics from the Ministry of Information Industry showed that there were more than 437 million mobile phone users in the Chinese mainland, or 327 mobile phones per 1, 000 population.
By the medieval Chinese dynasties, with the decline of chariot warfare, the use of the dagger-axe was almost nonexistent.
By 1954 an estimated 2, 500 Soviet naval advisers were in China — possibly one adviser to every thirty Chinese naval personnel — and the Soviet Union began providing modern ships.
By the end of the 17th century, the Chinese economy had recovered from the devastation caused by the wars in which the Ming Dynasty were overthrown, and the resulting breakdown of order.
By 28 BC, Chinese astronomers were regularly recording sunspot observations in official imperial records.
By then he had gained fluency in French, Chinese, Sanskrit, Pali, Japanese and English, in addition to his native Vietnamese.
By June 1598, The Japanese forces fought with desperation, turning back several Chinese offensives in Suncheon and Sacheon as the Ming army prepared for a final assault.
By 1920, Pan-Turkic Islamists had become a challenge to Chinese warlord Yang Zengxin ( 杨增新 ) who controlled Siankiang.
By 1900, restaurants serving Chinese cuisine from Canton and Shanghai offered a simple ramen dish of noodles ( cut rather than hand pulled ), a few toppings, and a broth flavored with salt and pork bones.
By and Remainder
By the Chinese Remainder Theorem, each can further be decomposed into a direct sum of submodules of the form, where is a power of a prime ideal.
By and Theorem
By Theorem 10, D is a diagonalizable operator which we shall call the diagonalizable part of T.
Therefore both and are inverses of By Theorem 1. 5,
By Fermat's Little Theorem, 2 < sup >( q − 1 )</ sup > ≡ 1 ( mod q ).
By the Hilbert Basis Theorem and some elementary properties of Noetherian rings, every affine or projective coordinate ring is Noetherian.
By applying the Coase Theorem two possible solutions arise for internalizing this externality.
By the First Fundamental Theorem, 0 ≤ δ ( a, f ) ≤ 1, if T ( r, f ) tends to infinity ( which is always the case for non-constant functions meromorphic in the plane ).
By Theorem 1. 3. 1 of Bekka et al., G is compactly generated.
By and ring
By Schur's lemma, is a division ring, and hence is a local ring, so right ( and left ) irreducible idempotents are local.
By 1994 the " 3174 Establishment Controller " supported features such as attachment to multiple hosts via token ring, ethernet, or X. 25 in addition to the standard channel attach or SDLC, and terminal attachment via twisted pair, token ring or ethernet in addition to co-ax.
* By connecting the computers at each end, a ring topology can be formed.
By the end of the American Civil War, the Observatory's clocks were linked via telegraph to ring the alarm bells in all of the Washington, D. C. firehouses three times a day, and by the early 1870s the Observatory's daily noon time signal was being distributed nationwide via the Western Union Telegraph Company.
By definition, all β-lactam antibiotics have a β-lactam ring in their structure.
By this the intensity of the emitted synchrotron radiation was increased a hundredfold in comparison to conventional storage ring systems.
By Stone's representation theorem every Boolean ring is isomorphic to a field of sets ( treated as a ring with these operations ).
By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.
By analyzing the bank accounts of certain members of the ring, he obtained legal proof of the principle on which the spoils had been divided.
By design there are more fingerings than humans have actual fingers, although both of these problems are solved by first fretting the initial string ( fourth on the downstroke ) with the tip of the ring finger, then rolling into the next string by fretting it with the pad of the same finger.
By combining a flat pick with metal finger picks on my middle and ring fingers, I discovered I could instantly switch from fast single-note runs to banjo rolls and get the best of both worlds.
By using the ring ropes to pull himself up, Taylor managed to return to his feet and was given the mandatory 8-count.
By this time a procedure for determining whether the ring of all algebraic integers of a given quadratic field is a PID was well known to the quadratic form theorists.
By then the sale of Vauxhall and its sister subsidiary, Opel, was being negotiated as part of a strategy driven by the German government to ring fence the businesses from any General Motors asset liquidation.
By adding together the external magnetic field and the induced field, the final configuration after compression can be obtained ; the total magnetic flux through the ring has been conserved ( even though the distribution of the magnetic flux has been modified ), and a current has been created in the conductive ring.
By the same mechanism, the signals of protons located near the ring axis are shifted up-field.
By twisting the ring, the symmetry of the system changes and becomes allowed ( see also Möbius – Hückel concept for details ).
By 1990, Fleiss parted with Alex and began to run her own prostitution ring.
1.095 seconds.