Problem 4. Let Rconsider the map:a b{(86) | a, b, c € Z}. Prove that R is a ring. Then0C=o: R ZOZ,a b*(( ! )).8 b )) = (a,c).0 cShow that is a ring homomorphism which is onto. Describe its kernel andshow it is a prime ideal.

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Answered: Problem 4. Let R = = consider the map:…

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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7. Let R be an integral domain. a) Show that if R is normal, then so is S-'R for any multiplicatively closed set S. b) Show that if the localization Rm at every maximal ideal m C R is normal, then R is normal. c) Show that the localization R. at an element a is isomorphic to the ring R[T]/(aT – 1), and interpret this statement geometrically. Is the assump- tion that R be an integral domain crucial for this statement?
Q6:Let (R, +..) and (R.+.) be commutative rings with identities. Given f is a homomorphism from (R, +..) onto (R, +.). Prove that if (I.+..) is a maximal ideal of (R, +..) then (f(1), +,) is a maximal ideal of (R,+.:).
(k) Let f: A B be a mapping of sets, A a system of subsets of A, and B a system of subsets of B. Define S(A) = {S(X) C B : X e A). (8) = {r-)eA :Y e B}. Prove that f-1(B) is a ring if B is a ring. (1) Show that f(A) is not necessarily a ring if A is a ring. (m) Show that -(B) is a a-algebra if B is a o-algebra. (n) Prove that R(s-(8) = (R(B). where R(C) is the minimal ring containing e.
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3. Determine whether the following mappings are ring homomorphisms: f: Q- Q defined by f(x)= |x| for all x e Q. a) b) c) g: CM₂ (R) defined by g (a + bi) = a b -b a h: Z[√2] → Z[√2] defined by h( a + b √2) =) a - b √2
Each of the quotient rings R/I on the left is isomorphic to one of the rings S on the right. Match each ring with its isomorphic partner, and prove that they really are isomorphic by constructing a surjective ring homomorphism p: RS with kernel I. R/I S A) Z[X]/(8, 12, X) 1. ZZ 2. Q 3. R B) Q[X]/(X² - 2) 4. C 5. F4 C) R[X]/(X-√2) 6. F8 7. ZA 8. Zg D) R[X]/(x²+X+2) 9. {a+b√2 a,bQ} 10. a {( 8 b ) | ab € F₁ } a E) R[X]/(X²) F) Z₁₂[X]/(X + X²+1) a 11. | a, b € F8 0 a 12. {( o b) | ab € R } a 0 a In examples 5, 6, 10, and 11 the symbols F4 and Fg denote a field of 4 and a field of 8 elements respectively. In examples 10, 11, and 12 the ring operations are addition and multiplication of matrices. Some methods you might use to try and match up R/I and S: (i) guess, (ii) think about representatives in the quotient ring, and what the multiplication rules are and try and match that up with something in the S column, (ii) try and make up a homomorphism from R to somewhere that would have elements…
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.
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part b and c solution needed
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Ring.Hom.
2. Assume that the set S I, y is a ring with respect to matrix addition and multiplication. D-r is a ring homomorphism from (a) Verify that the mapping e : S Z defined by e R to Z. (b) Find ker 0. (c) Find S/kere. (d) Find an isomorphism from S/kere to Z.

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Problem 4. Let R = = consider the map: a b {(86) | a, b, c € Z}. Prove that R is a ring. Then 0 C o: R ZOZ, * (()); b)) = (a,c). Show that is a ring homomorphism which is onto. Describe its kernel and show it is a prime ideal.