Problem 1. Let I and J be ideals in a ring commutative ring R with 1 R. Recall


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1 I. TakeHome Portion: Math 350 Final Exam Due by 5:00pm on Tues. 5/12/15 No resources/devices other than our class textbook and class notes/handouts may be used. You must work alone. Choose any 5 problems below to complete.* (*If you choose to answer all six problems, the sixth will count towards extra credit.) Problem 1. Let I and J be ideals in a ring commutative ring R with 1 R. Recall the sets IJ := {a 1 b 1 + a 2 b a n b n n 1, a k I, b k J}, I + J := {a + b a I, b J}. We have already shown that IJ and I + J are both ideals in R, so you make take this for granted. a) Suppose L is a prime ideal in a commutative ring R. Let I and J be ideals in R such that IJ L. Prove that either I L or J L. b) Now let L be any ideal in a commutative ring R (L is not necessarily prime). Let x, y R be such that xy L. Consider the ideals I = (x) + L and J = (y) + L. (Here, (w) denotes the principal ideal generated by the element w R.) Show that the ideal IJ satisfies IJ L. c) Prove the converse of part a). That is, suppose L is an ideal in a commutative ring R, L R, with the property that whenever two ideals I and J in R satisfy IJ L, either I L or J L. Prove that L is a prime ideal in R. Problem 2. Consider the ring R = (Z/3Z)[x]. a) Let r Z/3Z, and define the quotient ring T r = R/(x 2 + rx + 2). Determine all r Z/3Z for which the quotient ring T r is a field. Simply providing a list will not suffice; you must prove your answer. b) Choose any one of your fields T r from part a), and prove that T r = {ax + b + I a, b Z/3Z}. (Here, I = (x 2 + rx + 2) is the principal ideal used to define T r.) c) Using the same field T r from part b), determine the order T r of your field T r. 1
2 Problem 3. Let F be a field. a) Prove that F {0}, and {0} F are ideals in the ring R = F F. b) Prove that the four ideals F {0}, {0} F, {0} {0}, and F F are the only ideals in R. You may use without proof the fact that F F and {0} {0} are ideals in R. Hint. You may wish to start with the following observation: if there is some other ideal J in R, then J contains some element (a, b) (0, 0). Problem 4. Let U := 1 a b 0 1 c a, b, c F 2. Here, F 2 denotes the field of two elements (so F 2 = Z/2Z = {0, 1}). That is, U is the set of all 3 3 matrices with entries in F 2 that have 1s along the diagonal, and 0s below the diagonal. a) Prove that U is a group. b) Let H := {I, I, i, i, j, j, k, k} be the quaternion group, let D 4 be the dihedral group of degree 4, and let Z := Z/2Z Z/4Z. The group U is either isomorphic to H, D 4, or Z. Determine which of these three groups U is isomorphic to, and then prove the isomorphism. Problem 5. Let G be a group, and let H G be a subgroup. Let S := {gh g G} be the set of all left cosets of H in G. Let a G and define the map f a : S S by f a (gh) = agh. a) Prove that f a is well defined, and is a bijection from S to S. b) Let A(S) := {f : S S f is a bijection} denote the permutation group of the set S. We have already shown for any set T that A(T ) is a group, so you may take this for granted. Prove that the map φ : G A(S) defined by φ(a) = f a is a group homomorphism. Problem 6. a) Let n 3 be an integer. Prove that any kcycle σ in S n, where k is odd, is such that σ 2 is also a cycle. b) Let n 4 be an integer. Prove that any kcycle σ in S n, where k 4 is even, is such that σ 2 is not a cycle. 2
3 c) Let n 2 be an integer. Prove that if σ = (a 1 a 2... a k ) is a kcycle in S n, then for all r {1, 2, 3,..., k}, we have that { σ r a j+r, if j + r k, (a j ) = a j+r, if j + r > k. Here, for an integer m we let m denote its least positive residue (mod k). For example, if k = 7, we have 8 = 1, 12 = 2, 7 = 7, etc. d) Deduce from part c) that if σ is a kcycle in S n that σ = k. (You may not simply state this as a fact.) Hint. You may find it helpful to look at some explicit examples involving 3cycles, 4cycles, etc. before you begin the general proofs. 3 Scheduled Portion begins on next page
4 II. Scheduled Portion: Math 350 Final Exam Mon. 5/11/15 2pm5pm, SMUD 205 No external resources/devices may be used. Choose any 5 problems below to complete.* (*If you choose to answer all six problems, the sixth will count towards extra credit.) Problem 1. For each item below, decide if the statement is true or false. If true, provide a proof. If false, provide a counterexample or disprove. a) T / F There exists a nontrivial group homomorphism φ : Z/6Z S 3. Recall a group homomorphism φ : G H is called trivial if φ(g) = e H for all g G. b) T / F Let G be a group, let N G a normal subgroup, and let a G. Then a in G is equal to Na in the quotient group G/N. c) T / F If φ : R S is a ring homomorphism, then im(φ) := {φ(r) r R} is an ideal in S. Problem 2 Let I and J be ideals in a ring R, and let φ : R R/I R/J be defined by φ(r) = (r + I, r + J) (r R). a) Show that φ is a ring homomorphism. b) Determine ker(φ). c) Let R = Z, I = 4Z, and J = 2Z. In this case, is φ surjective? Prove or disprove. 4
5 Problem 3. Let φ : R S be a homomorphism of rings. a) Prove that ker(φ) = {0} if and only if φ is injective. b) Suppose φ : R S is ring homomorphism (i.e. S is a ring). Prove that either im(φ) = R or im(φ) = {0}. Recall that if φ : R S is a ring homomorphism, the set im(φ) := {φ(r) r R}. Problem 4. Let φ : G K be a group homomorphism. Let M K be a normal subgroup in K. Define the set N = {g G φ(g) M}. a) Prove that N G is a normal subgroup in G. b) Assume φ : G K is surjective. Define the map ψ : G K/M by ψ(g) = Mφ(g) (where g G). Prove that ψ is a surjective group homomorphism. Problem 5. a) Let α and β be any two transpositions in S n (not necessarily distinct). (Recall a transposition is also called a 2cycle. ) Prove that the element βαβ 1 is also a transposition. b) Now let α S n be a transposition, and let β be any element in S n. Prove that βαβ 1 is also a transposition. Problem 6. Let φ : D 10 H be a group homomorphism, where D 10 is the dihedral group of degree 10 and order 20, and H is a finite group of order 63. Prove that φ must be the trivial homomorphism. Recall a group homomorphism φ : G H is called trivial if φ(g) = e H for all g G. 5
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