Abstract Algebra Theory and Applications - Computer Science ...

252 CHAPTER 14 RINGS11. Prove that the Gaussian integers, Z[i], are an integral domain.12. Prove that Z[ √ 3 i] = {a + b √ 3 i : a, b ∈ Z} is an integral domain.13. Solve each of the following systems of congruences.(a) x ≡ 2 (mod 5)x ≡ 6 (mod 11)(b) x ≡ 3 (mod 7)x ≡ 0 (mod 8)x ≡ 5 (mod 15)(c) x ≡ 2 (mod 4)x ≡ 4 (mod 7)x ≡ 7 (mod 9)x ≡ 5 (mod 11)(d) x ≡ 3 (mod 5)x ≡ 0 (mod 8)x ≡ 1 (mod 11)x ≡ 5 (mod 13)14. Use the method of parallel computation outlined in the text to calculate2234 + 4121 by dividing the calculation into four separate additions modulo95, 97, 98, and 99.15. Explain why the method of parallel computation outlined in the text failsfor 2134 · 1531 if we attempt to break the calculation down into two smallercalculations modulo 98 and 99.16. If R is a field, show that the only two ideals of R are {0} and R itself.17. Let a be any element in a ring R with identity. Show that (−1)a = −a.18. Prove that (−a)(−b) = ab for any elements a and b in a ring R.19. Let φ : R → S be a ring homomorphism. Prove each of the following statements.(a) If R is a commutative ring, then φ(R) is a commutative ring.(b) φ(0) = 0.(c) Let 1 R and 1 S be the identities for R and S, respectively. If φ is onto,then φ(1 R ) = 1 S .(d) If R is a field and φ(R) ≠ 0, then φ(R) is a field.20. Prove that the associative law for multiplication and the distributive lawshold in R/I.21. Prove the Second Isomorphism Theorem for rings: Let I be a subring of aring R and J an ideal in R. Then I ∩ J is an ideal in I andI/I ∩ J ∼ = I + J/J.