16.1.12. Let I be the ideal generated by 3 in Z/36Z. Show that as a group I isisomorphic to (Z/12Z, +). Show that the ring I is not isomorphic to thering (Z/12Z, +,). Similarly show that in Z/36Z, (6) is not isomorphic asa ring to Z/6Z.

Answered: 16.1.12. Let I be the ideal generated…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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2
6. Let S be the set of 2 × 2 matrices over Z of the form (?). Show that (a) S is a ring. (b) (a b)* = (a* .*). 0 0 where denotes some integer. Also find the idempotents and nilpotent elements of S. Show that nilpotent elements form a subring.
Answer 16 and 17
15. (a) Show that S= {(a,a) | a e Z} is a subring of ZxZ. (Use the definition of addition and multiplication of direct products.) (b) Determine if T = {(a,-a) | ae Z} is a subring of ZxZ. 16. Let R be a commutative ring with unity and let
5. (*) Let R be a ring with 1 in which ab 0 whenever a, b E R are nonzero. Let a ER\ {0}. An element b E R is said to be a left (respectively, right) inverse for a if ba = 1R (respectively, ab = 1R). Show that if a E R has a left inverse b then b is also a right inverse for a (and thus is an inverse for a).
4. (a) Let R = {(: ) | a, b,c e Z}. Given that R is a ring under the usual matrix addition and multiplication operations, prove that x) E R U = is an ideal of the ring R. (b) Using Fermat's Little Theorem, find 7188 mod 13. (c) Let K {a + bi e C : a, b E Z}. (i) State whether K, together with the usual complex number addition and multiplication operat is a Euclidean domain; and if so, give a Euclidean function for K. (ii) Determine q, r € K such that 6 + 4i = (2 – 3i)q +r.
9. Iff js a homomorphism from a ring R into a ring S then show that f(0)=0 1 :: f(- a) = -f(a), V a e R.
9. The number of ideals in the ring (Z1s, +is15) is: (a) Two (b) Three (e) Four (d) Five.
17. Given that (I, 1.) in an ideal of the ring (R, +,), show that a) whenever (R, 1,) is commutative with identity, then so is the quotiont ring (R/1,+,), b) the ring (R/1,+,) may have divisors of zero, even though (R, +,) docs not have any, c) if (R,+,) is a principal ideal ring, then so is the quotient ring (R/I, +,).
It is not always possible to construct by ruler and compass, a square equal in area to the area of a circle.
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9. a) find two integers x,y such that 30x+101y=1. (hint: gcd(30,101)=1). b) find the inverse of 30 in the group U(101).

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16.1.12. Let I be the ideal generated by 3 in Z/36Z. Show that as a group I is isomorphic to (Z/12Z, +). Show that the ring I is not isomorphic to the ring (Z/12Z, +,). Similarly show that in Z/36Z, (6) is not isomorphic as a ring to Z/6Z.