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.Therefore,1 + M = ab + M = ba + M = (a + M )(b + M ).Conversely, suppose that M is an ideal and R/M is a field.Since R/Mis a field, it must contain at least two elements: 0 + M = M and 1 + M.Hence, M is a proper ideal of R.Let I be any ideal properly containing M.We need to show that I = R.Choose a in I but not in M.Since a + M is anonzero element in a field, there exists an element b + M in R/M such that(a + M )(b + M ) = ab + M = 1 + M.Consequently, there exists an elementm ∈ M such that ab + m = 1 and 1 is in I.Therefore, r1 = r ∈ I for allr ∈ R.Consequently, I = R.Example 18.Let pZ be an ideal in Z, where p is prime.Then pZ is amaximal ideal since∼Z/pZ = Zp is a field.An ideal P in a commutative ring R is called a prime ideal if wheneverab ∈ P , then either a ∈ P or b ∈ P.Example 19.It is easy to check that the set P = {0, 2, 4, 6, 8, 10} is anideal in Z12.This ideal is prime.In fact, it is a maximal ideal.Proposition 14.16 Let R be a commutative ring with identity.Then P isa prime ideal in R if and only if R/P is an integral domain.14.4MAXIMAL AND PRIME IDEALS245Proof.First let us assume that P is an ideal in R and R/P is an integraldomain.Suppose that ab ∈ P.If a + P and b + P are two elements of R/Psuch that (a + P )(b + P ) = 0 + P = P , then either a + P = P or b + P = P.This means that either a is in P or b is in P , which shows that P must beprime.Conversely, suppose that P is prime and(a + P )(b + P ) = ab + P = 0 + P = P.Then ab ∈ P.If a /∈ P , then b must be in P by the definition of a primeideal; hence, b + P = 0 + P and R/P is an integral domain.Example 20.Every ideal in∼Z is of the form nZ.The factor ring Z/nZ = Znis an integral domain only when n is prime.It is actually a field.Hence, thenonzero prime ideals in Z are the ideals pZ, where p is prime.This examplereally justifies the use of the word “prime” in our definition of prime ideals.Since every field is an integral domain, we have the following corollary.Corollary 14.17 Every maximal ideal in a commutative ring with identityis also a prime ideal.Historical NoteAmalie Emmy Noether, one of the outstanding mathematicians of this century, wasborn in Erlangen, Germany in 1882.She was the daughter of Max Noether (1844–1921), a distinguished mathematician at the University of Erlangen.Together withPaul Gordon (1837–1912), Emmy Noether’s father strongly influenced her earlyeducation.She entered the University of Erlangen at the age of 18.Althoughwomen had been admitted to universities in England, France, and Italy for decades,there was great resistance to their presence at universities in Germany.Noetherwas one of only two women among the university’s 986 students.After completingher doctorate under Gordon in 1907, she continued to do research at Erlangen,occasionally lecturing when her father was ill.Noether went to Göttingen to study in 1916.David Hilbert and Felix Kleintried unsuccessfully to secure her an appointment at Göttingen.Some of the facultyobjected to women lecturers, saying, “What will our soldiers think when they returnto the university and are expected to learn at the feet of a woman?” Hilbert,annoyed at the question, responded, “Meine Herren, I do not see that the sex ofa candidate is an argument against her admission as a Privatdozent.After all,246CHAPTER 14RINGSthe Senate is not a bathhouse.” At the end of World War I, attitudes changedand conditions greatly improved for women.After Noether passed her habilitationexamination in 1919, she was given a title and was paid a small sum for her lectures.In 1922, Noether became a Privatdozent at Göttingen.Over the next 11 yearsshe used axiomatic methods to develop an abstract theory of rings and ideals.Though she was not good at lecturing, Noether was an inspiring teacher.One of hermany students was B.L.van der Waerden, author of the first text treating abstractalgebra from a modern point of view.Some of the other mathematicians Noetherinfluenced or closely worked with were Alexandroff, Artin, Brauer, Courant, Hasse,Hopf, Pontryagin, von Neumann, and Weyl.One of the high points of her careerwas an invitation to address the International Congress of Mathematicians in Zurichin 1932.In spite of all the recognition she received from her colleagues, Noether’sabilities were never recognized as they should have been during her lifetime.Shewas never promoted to full professor by the Prussian academic bureaucracy.In 1933, Noether, a Jew, was banned from participation in all academic activi-ties in Germany.She emigrated to the United States, took a position at Bryn MawrCollege, and became a member of the Institute for Advanced Study at Princeton.Noether died suddenly on April 14, 1935.After her death she was eulogized bysuch notable scientists as Albert Einstein.14.5An Application to Software DesignThe Chinese Remainder Theorem is a result from elementary number theoryabout the solution of systems of simultaneous congruences.The Chinesemathematician Sun-ts¨ı wrote about the theorem in the first century A.D.This theorem has some interesting consequences in the design of softwarefor parallel processors.Lemma 14.18 Let m and n be positive integers such that gcd(m, n) = 1.Then for a, b ∈ Z the systemx≡ a(mod m)x≡ b(mod n)has a solution.If x1 and x2 are two solutions of the system, then x1 ≡ x2(mod mn).Proof.The equation x ≡ a (mod m) has a solution since a + km satisfiesthe equation for all k ∈ Z.We must show that there exists an integer k1such thata + k1m ≡ b(mod n).14.5AN APPLICATION TO SOFTWARE DESIGN247This is equivalent to showing thatk1m ≡ (b − a)(mod n)has a solution for k1.Since m and n are relatively prime, there exist integerss and t such that ms + nt = 1 [ Pobierz całość w formacie PDF ]