Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain. Prove that I (J +K) = I J +I K. Problem 7. Let R be a commutative ring and I be a proper prime ideal of R such that R/I satisfies the descending chain condition on ideals. Prove that R/I is a field. Problem 8. Let R be a commutative ring and I be an ideal which is contained in a prime ideal P. Prove that the collection of prime ideals contained in P and containing I has a minimal member. Problem 9. Let X be a finite set and let R be the ring of functions from X into the field R of real numbers. Prove that an ideal M of R is maximal if and only if there is an element a X such that M = {f f R and f(a) = 0}. Algebra Homework, Edition 3 Due 16 Septemer 2010 Problem 10. Let R be a commutative ring and let n be a positive integer. Let J, I 0, I 1,..., I n 1 be ideals of R so that I k is a prime ideal for every k < n and so that J I 0 I n 1. Prove that J I k for some k < n. Problem 11. Let R be a nontrivial commutative ring and let J be the intersection of all the maximal proper ideals of R. Prove that 1 + a is a unit of R for all a J.
2 Algebra Homework, Edition 4 Due 23 September 2010 Problem 12. Let F be a field and let p(x) F[x] be a polynomial of degree n. Prove that p(x) has at most n distinct roots in F. Problem 13. Let F be a field and let F be its (multiplicative) group of nonzero elements. Let G be any finite subgroup of F. Prove that G must be cyclic. Problem 14. Suppose that D is a commutative ring such that D[x] is a principal ideal domain. Prove that D is a field. Algebra Homework, Edition 5 Due 30 September 2010 Problem 15. Is the polynomial y 3 x 2 y 2 + x 3 y + x + x 4 irreducible in Z[x, y]? Problem 16. Let R be a principal ideal domain, and let I and J be ideals of R. IJ denotes the ideal of R generated by the set of all elements of the form ab where a I and b J. Prove that if I + J = R, then I J = IJ. Problem 17. Let D be a unique factorization domain and let I be a nonzero prime ideal of D[x] which is minimal among all the nonzero prime ideals of D[x]. Prove that I is a principal ideal.
Algebra Homework, Edition 6 Due 7 October 2010 3 Problem 18. a. Prove that (2, x) is not a principal ideal of Z[x]. b. Prove that (3) is a prime ideal of Z[x] that is not a maximal ideal of Z[x]. Problem 19. Show that any integral domain satisfying the descending chain condition on ideals is a field. Problem 20. Prove the following form of the Chinese Remainder Theorem: Let R be a commutative ring with unit 1 and suppose that I and J are ideals of R such that I + J = R. Then R I J = R I R J. Algebra Homework, Edition 7 Due 21 October 2010 Problem 21. Let D be an integral domain and let c 0,..., c n 1 be n distinct elements of D. Further let d 0,..., d n 1 be arbitrary elements of D. Prove there is at most one polynomial f(x) D[x] of degree n 1 such that f(c i ) = d i for all i < n. Problem 22. Let F be a field and let c 0,..., c n 1 be n distinct elements of F. Further let d 0,..., d n 1 be arbitrary elements of F. Prove there is at least one polynomial f(x) F [x] of degree at most n 1 such that f(c i ) = d i for all i < n. Problem 23. Let R be the following subring of the field of rational functions in 3 variables with complex coefficients: { } f R = : f, g C[x, y, z] and g(1, 2, 3) 0 g Find 3 prime ideals P 1, P 2, and P 3 in R with 0 P 1 P 2 P 3 R.
4 Algebra Homework, Edition 8 Due 28 October 2010 Problem 24. Prove that the polynomial x 3 y + x 2 y xy 2 + x 3 + y is irreducible in Z[x, y]. Problem 25. Let D be a subring of the field F. An element r F is said to be integral over D provided there is a monic polynomial f(x) D[x] such that r is a root of f(x). For example, the real number 2 is integral over the ring of integers since it is a root of x 2 2. Now suppose D is a unique factorization domain and F is its field of fractions. Prove that the set of elements of F that are integral over D coincides with D itself. Problem 26. Prove that there is a polynomial f(x) R[x] such that (a) f(x) 1 belongs to the ideal (x 2 2x + 1); (b) f(x) 2 belongs to the ideal (x + 1), and (c) f(x) 3 belongs to the ideal (x 2 9). Algebra Homework, Edition 9 Due 4 November 2010 Problem 27. Let A be the 4 4 real matrix (a) Determine the rational canonical form of A. (b) Determine the Jordan canonical form of A. 1 1 0 0 A = 1 1 0 0 2 2 2 1 1 1 1 0 Problem 28. Suppose that N is a 4 4 nilpotent matrix over a field F with minimal polynomial x 2. What are the possible rational canonical forms for N?
Algebra Homework, Edition 10 Due 16 November 2010 5 Problem 29. Let R be a nontrivial integral domain and let M be an R-module. elements is a submodule of M and M/T is torsion free. Then the set T of torsion Problem 30. Let R be a principal ideal domain and let T be a torsion R-module of exponent r. Then T has an element of order r. Problem 31. Prove that the sequence of invariant factors (i.e. the sequence r 0, r 1,..., r n ) mentioned in the Invariant Factor Theorem is uniquely determined by the module. Problem 32. Let M be a finitely generated R-module, where R is a principal ideal domain. Prove each of the following. (a) The direct decomposition of M using the Invariant Factor Theorem is the one using the smallest number of direct factors that are all cyclic. (b) The direct decomposition of M using the Elementary Divisor Theorem is the one using the largest number of nontrivial direct factors that are all cyclic. Algebra Homework, Edition 11 Due 2 December 2010 Problem 33. Let R be a commutative ring and let S be a subring of R so that S is Noetherian. Let a R and let S be the subring of R generated by S {a}. Prove that S is Noetherian. Problem 34. Let R be a commutative ring. An R-module P is said to be projective provided for all R-modules M and N and all homomorphisms f from M onto N, if g is a homomorphism P into N, then there is a homomorphism h from P into M so that f h = g. Prove that every free R-module is projective. Problem 35. Let R by a principal ideal domain and let M and N be finitely generated R-modules such that M M = N N. Prove M = N. Problem 36. Give an example of two 4 4 matrices with real entries that have the same minimal polynomial and the same characteristic polynomial but are not similar.