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Math 6310, Fall 2017 Homework 8 1. Let I and J Be Comaximal 2-Sided

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Math 6310, Fall 2017 Homework 8 1. Let I and J Be Comaximal 2-Sided Math 6310, Fall 2017 Homework 8 1. Let I and J be comaximal 2-sided ideals of a ring R. Show that I \ J = IJ + JI. + + 2. The Euler function φ : Z ! Z is defined by ¯ φ(n) := jfi 2 Zn j gcd(i; n) = 1gj: × (a) Show that φ(n) = jZn j. (b) Show that if p is prime then φ(pa) = pa − pa−1. (c) Show that if gcd(ni; nj) = 1 for all i 6= j then φ(n1 ··· nk) = φ(n1) ··· φ(nk). a1 ak (d) Write n = p1 ··· pk where the pi are distinct primes. Show that 1 1 φ(n) = n(1 − ) ··· (1 − ): p1 pk + φ(n) 3. (a) Show that for any n 2 Z and a 2 Z with gcd(a; n) = 1, we have a ≡ 1 mod n. p (b) Show that for any prime p and a 2 Z, we have a ≡ a mod p (Fermat's little theorem). [0;1] 4. Let R := R be the ring of all functions [0; 1] ! R, and S := C[0; 1] the subring of continuous functions. (a) Show that neither R nor S are noetherian by constructing a strictly increasing chain of ideals in each. (b) Let I := ff 2 R j f(1=2) = 0g. Show that I is a principal ideal of R. (c) Let J := ff 2 S j f(1=2) = 0g. Show that the ideal J of S is not finitely generated and deduce again that S is not noetherian. [Suppose J = (f1; : : : ; fk), define g := maxfjf1j;:::; jfkjg. Show that: (i) there is a sequence x ! 1=2 such that g(x ) 6= 0 for all n; (ii) there is a constant M > 0 p n n such that g ≤ M · g.] ∼ ∼ (d) Show that R=I = R = S=J and deduce that I and J are maximal (in R and S, respectively). 5.( Field of fractions) Let D be an integral domain and D] := D n f0g. Define operations + and · on D × D as follows: (a; b) + (c; d) := (ad + bc; bd); (a; b) · (c; d) := (ac; bd): Note that each operation turns D × D into a monoid and that D × D] is a submonoid. (a) Define a relation on D × D] as follows: (a; s) ∼ (a0; s0) if as0 = a0s: Show that ∼ is an equivalence relation and that it is compatible with + and ·. 1 a ] ] (b) Let s denote the class of (a; s) 2 D × D and Q := (D × D )= ∼ denote the set of equivalence classes. Show that Q is a field with operations a b at + bs a b ab + := ; · := : s t st s t st Make sure you check the distributivity axiom, among other things. a (c) Let δ : D ! Q be δ(a) = 1 . Show that δ is an injective morphism of rings. (d) Let F be a field and ' : D ! F an injective morphism of rings. Show that there is a unique morphism of fields 'b : Q ! F such that the diagram below commutes. D δ / Q ' ' b F Deduce that Q is isomorphic to a subfield of F . We say that Q is the field of fractions of D. 6.( Ring of fractions) Let R be a commutative ring. A subset of R is multiplicatively closed if it is a submonoid of (R; ·; 1). Let S be such a subset. (a) Define a relation on R × S as follows: (a; s) ∼ (a0; s0) if there is t 2 S such that as0t = a0st. Show that ∼ is an equivalence relation and that it is compatible with + and ·. (These operations on R × S are defined as in Exercise5.) a −1 (b) Let s denote the class of (a; s) 2 R × S and S R denote the set of equivalence classes. Show that S−1R is a commutative ring with operations a b at + bs a b ab + := ; · := : s t st s t st Make sure you check the distributivity axiom, among other things. (c) Show that S−1R is the trivial ring if and only if 0 2 S. −1 a (d) Let δ : R ! S R be δ(a) = 1 . Show that δ is a morphism of rings. (e) Let ' : R ! R0 be a morphism of rings such that '(s) is invertible in R0 for each −1 0 s 2 S. Show that there is a unique morphism of rings 'b : S R ! R such that the diagram below commutes. R δ / S−1R ' ' b " R0 We say that S−1R is the ring of fractions of R with denominators in S. 7. Let R be a commutative ring and I an ideal of R. We say that I is prime if given a; b 2 R, if ab 2 I, then either a 2 I or b 2 I. 2 (a) Show that I is prime if and only if R=I is an integral domain. (b) Show that R is an integral domain if and only if f0g is a prime ideal. (c) Show that if I is maximal, then it is prime. (d) Let f : R ! R0 be a morphism of rings and J a prime ideal of R0. Show that f −1(J) is a prime ideal of R. 8. Let R be a commutative ring. Suppose there exists a prime ideal P of R which doesn't contain any zero divisor. Show that R is an integral domain. 9. Let R be a commutative ring. (a) Let I be an ideal and S = R n I. Show that I is prime and proper if and only if S is multiplicatively closed. The ring of fractions S−1R, with S = R n I, is the localization of R at I. n (b) Let a be an element of R and S = fa j n 2 Ng. Show that S is multiplicatively closed and that S−1R is the trivial ring if and only if a is nilpotent. (An element n a 2 R is nilpotent if there is n 2 N such that a = 0.) (c) Let S = Rnf0g. Show that R is an integral domain if and only if S is multiplicatively closed. Show that in this case the equivalence relation of Exercise6 reduces to that of Exercise5. (Hence, the ring of fractions is in this case the field of fractions.) 10. Let R be a nontrivial ring, D a division ring, and ' : D ! R a morphism of rings. Show that ' is injective. 11. Let G be a group and | a commutative ring. Let |G be the group algebra of G, as in Homework 7. −1 (a) Given x; g 2 G and f 2 |G, show that (f ∗ δg)(x) = f(xg ). (b) A function f : G ! | that is constant on each conjugacy class of G is called a class function. Show that Z(|G) is precisely the subspace of class functions. [Consider δg−1 ∗ f ∗ δg.] 3 12. (a) Let R be a left noetherian ring and let S be an arbitrary subset of R. Show that there is a finite subset S0 ⊆ S such that hSi = hS0i. (b) Let | be a commutative noetherian ring. Given a set S ⊆ |[x1; : : : ; xn], let n Z(S) := fa 2 | j f(a) = 0 for all f 2 Sg: Show that there is a finite subset S0 ⊆ S such that Z(S) = Z(S0). The first part is a strong version of the result seen in class that all left ideals in a left noetherian ring are finitely generated. The second states that given a system of polynomial equations, we can always find finitely many among them such that the cor- responding subsystem has the same solutions. 13. A ring is left artinian ring if it satisfies the descending chain condition on left ideals, or equivalently (why?) if every nonempty set of left ideals has a minimal element. (a) Show that Z is not artinian. (b) Let | be a field A be a |-algebra such that dim| A < 1. Show that A is left artinian and left noetherian. (Recall our rings are always unital.) It is a consequence of the Akizuki-Hopkins-Levitzki theorem that a left artinian ring is necessarily left noetherian. Part (a) shows that the converse does not hold. Also, the corresponding statement for modules does not hold. 14. Let I be a left ideal of R and M a left R-module. Let IM denote the set of finite linear combinations of elements of M with coefficients in I. (a) Show that IM is an R-submodule of M. (b) Suppose that the ideal I is 2-sided. (i) Make M=IM into a left R=I-module. (ii) Let N be a left R=I-module. Make N into an R-module with IN = 0. 15. Let R be a commutative ring in which all prime ideals are finitely generated. This exercise will prove a theorem of I. S. Cohen stating that such a ring is noetherian. Suppose R has an ideal which is not finitely generated. (a) Show that there is an ideal I which is maximal among those which are not finitely generated. (b) Show that R=I is noetherian. (c) Find ideals J and K such that I ⊂ J; I ⊂ K yet JK ⊆ I: (Note the proper inclusions on the left.) [Use the hypothesis on prime ideals!] (d) Show that J, K and JK are finitely generated over R. (e) Show that J=JK is a module over R=I, and as such it is finitely generated. (f) Deduce that I=JK is finitely generated over R=I, hence over R.
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