Sma2110:Real Analysis Question Paper
Sma2110:Real Analysis
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2012
QUESTION ONE (30 MARKS)
(a) Let R be ring with 8 elements consisting of all 3 × 3 matrices with entries in 2 which have the following form.
Assume the standard laws for matrix addition and multiplication hold. (i) Show that R is a commutative ring. (3 Marks) (ii) Find all units of R and all mipotent elements of R. (3 Marks) (iii) Find the idempotent elements of R. (3 Marks)
(b) Let R be the ring 2/< 2 + 1 >. Show that although R has 4 elements, it is not isomorphic to either rings 4 2 2. (5 Marks)
(c) Find all maximal ideals and all prime ideals of 36 = 36 (5 Marks)
(d) Let R be a ring and let I and J be ideals in R. Which of the following is an ideal in R.
(i) + = + | , (2 Marks)
(ii) (2 Marks)
2
(iii) (2 Marks) Give a counter example whenever it is not an ideal. (e) Which of the following is not a unit in (i) (ii) 1 (iii) + 1 (iv) (1 Mark)
(f) Show that R is commutative if and only if + 2 = 2 + 2 + 2 (4 Marks)
QUESTION TWO (20 MARKS)
(a) Prove that the centre of a ring is a subring (3 Marks)
(b) Let =
Show that X is a subring of 2 . (2 Marks)
(c) Let be the subring of the field of complex numbers given by = + + 2 + 2 mean addition modulo 2. (i) Show that is a ring homomorphism. (5 Marks) (ii) Find Ker . (3 Marks) (iii) Show that Ker is a principal ideal of . (3 Marks) (d) Let a, b be commutative elements of a ring R of characteristic zero. Show that + 2 = 2 + 2 = 2. (2 Marks)
QUESTION THREE (20 MARKS)
(a) Prove that in an integral domain, the cancellation laws hold. (3 Marks)
(b) Consider the ring: = + 17 , verify that (i) 9 217 is prime. (5 Marks)
(ii) 15 + 717 is reduable (5 Marks)
(c) Prove that the characteristic of an integral domain is either zero or a prime. (3 Marks)
3
(d) Show that the ring 2 of even integers contain a maximal ideal such that 2/ is not a field. (4 Marks)
QUESTION FOUR (20 MARKS)
(a) Find the greatest common division of = 33 + 32 + 3 + 3 and give = 3 + 3 over expressed in the form = + () . (5 Marks)
(b) Prove that the polynomial ring over the filed F is a Euclidean ring. (5 Marks) (c) Prove or disapprove the following statement. The ring is a principal ideal domain. (4 Marks) (d) Prove that (i) 2 + + 1 is irreducible in 2 (3 Marks) (ii) 2 + 1 is irreducible in 7 (3 Marks)
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