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LANCASTER UNIVERSITY
2011 EXAMINATIONS
PART II (Second Year) MATHEMATICS & STATISTICS 2 hours Math 225: Groups and Rings
You should answer ALL Section A questions and THREE Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. SECTION A
A1. (a) State the definition of a group.^ [3] (b) Let S = Z \ {− 2 }. Show that a ∗ b = (a + 2)(b + 2) − 2 defines a binary operation on S, and with this operation, S is a group. [8]
A2. Let H and K be subgroups of a group G. Show that H ∩ K is also a subgroup of G. [3]
A3. Define a normal subgroup of a group G. [2] Show that the centre Z(G) = {z ∈ G : zg = gz ∀ g ∈ G} is a normal subgroup of G. [4]
A4. Decide whether each of the following statements is true or false. Briefly justify your answer. [10] (a) There exist no elements of Sn of order greater than n. (b) Any subgroup of an abelian group is abelian. (c) Any group G is also a ring. (d) ̂7 is the multiplicative inverse of 10 in̂ Z 23. (e) The subset
a b c d
∈ M 2 (R) : ad − bc = 0
of invertible 2 × 2 matrices with real coefficients is a subring of M 2 (R).
please turn over
SECTION A
A5. (^) (a) State the definition of a maximal ideal in a ring. [2] (b) Suppose that R is a commutative ring with a multiplicative identity. Find the unique true statement among the following ones. Justify your claim. (i) Any prime ideal in R is also maximal. (ii) Any maximal ideal in R is also prime. (iii) None of the above is true. (^) [5]
A6. Consider the ring R = Z[X] and let I = {f (X) ∈ R : f (X) = 9g(X) + (X^2 − 1)h(X) for some g(X), h(X) ∈ R} be the ideal of R generated by the elements 9 and X^2 − 1. (a) Find all the ideals of R which contain I. [8] (b) For every maximal ideal J of R containing I, use the Fundamental Isomorphism Theorem for rings to prove that the quotient ring R/J is isomorphic to Z 3. [5]
please turn over
SECTION B
B3. Let R be a unital commutative ring and A an ideal of R. Define N = {x ∈ R : xn^ = 0 for some n ∈ N } and N (A) = {x ∈ R : xn^ ∈ A for some n ∈ N }. (a) Show that N = N ({ 0 }). [1] (b) Show that N (A) is an ideal of R, and that N (A) contains A. [5] (c) Show that N (N (A)) = N (A). You may use the result stated in (b). [4] (d) Let R = Z 27 and A = {̂ 9 ̂k : ̂k ∈ Z 27 } = ̂ 9 R. Find N (A) and show that the quotient ring R/N (A) is a field. [10]
B4. (a) Let m ∈ { 2 , 3 , 4 ,.. .}. (i) Find all the surjective ring homomorphisms Z → Zm. [4] (ii) Find all the ring homomorphisms Zm → Z. [3] (b) Let G be an abelian group, and let E be the set of all group homomorphisms G → G. Define an addition and a multiplication on E as follows. For any ϕ, ψ ∈ E, the maps (ϕ + ψ), (ϕψ) ∈ E are defined by the rules: for all g ∈ G, { (^) (ϕ + ψ)(g) = ϕ(g) + ψ(g) (pointwise addition) (ϕψ)(g) = (ϕ ◦ ψ)(g) = ϕ(ψ(g))^ (composition) (i) Prove that E is a ring. [9] (ii) Prove that E is unital. [1] (iii) Is E necessarily a commutative ring? Briefly justify your answer. [3]
end of exam
