Let Z(2^(1/2)) denote the set {a+ b(2^(1/2))l a, b are integers}. Show that Z(2^(1/2)) is a subring of the all reals. #13 on the image. Thomas W. Hungerford - Abstrac x> Show that Z(\12) is a subring of Xb MATLAB: An Introduction with A X+O File | C:/Users/angel/Downloads/Thomas%20W.%20Hungerford%20-%20Abstract%20Algebra_%20AN%20lntroduction-Cengage%20Learning%20(2014).pdf...Flash Player will no longer be supported after December 2020.Turn offLearn more+-- A' Read aloudV DrawF HighlightO Erase77of 621(a) Prove that S is a ring.(b) Show that J =sa right identity in S (meaning that AJ = A forevery A in S).(c) Show that Jis not a left identity in S by finding a matrix B in S such thatJB + B.For more information about S, see Exercise 41.12. Let Z[] denote the set {a + bi |a, bɛZ}. Show that Z[] is a subring of C.13. Let ZV2] denote the set {a + bV2|a, b€Z}. Show that Z[V2] is a subringof R. See Example 20.]14. Let T be the ring in Example 8. Let S = {ƒeT|f(2) = 0}. Prove that S is asubring of T.15. Write out the addition and multiplication tables for(a) Zz x Z,(b) Z, x Z2(c) Zz x Z,16. Let A =0 =in M(R). Let S be the set of all matrices Bsuch that AB = 0.(a) List three matrices in S. [Many correct answers are possible.](b) Prove that S'is a subring of M(R). [Hìnt: If B and Care in S, show thatB+ C and BC are in S by computing A(B + C) and A(BC).]17. Define a new multiplication in Z by the rule: ab = 0 for all a, b,ɛZ Show thatwith ordinary addition and this new multiplication, Z is a commutative ring.18. Define a new multiplication in Z by the rule: ab = 1 for all a, b, eZ. Withordinary addition and this new multiplication, is Z is a ring?19. Let S= {a, b, c} and let P(S) be the set of all subsets of S; denote theelements of P(S) as follows:S = {a, b, c}; D = {a, b}; E= {a, c}; F= {b, c};A= {a}; B- {b}; C= {c}; 0-Ø.Define addition and multiplication in P(S) by these rules:M +N= (M – N)U (N – M)andMN = MO N.Write out the addition and multiplication tables for P(S). Also, see Exercise 44.B. 20. Show that the subset R = {0, 3, 6, 9, 12, 15} of Zg is a subring. Does R havean identity?21. Show that the subset S = {0, 2, 4, 6, 8} of Z10 is a subring. Does S have anidentity?

Test for the Subring: The non empty set S is subring of a ring R if ∀a,b∈S a-b∈Sa·b∈S

Answered: 13. Let ZV2denote the set (a + V2|a,…

13. Let ZV2denote the set (a + V2|a, beZ}. Show that ZV2a subring of RSee Example 20.] 14. Let Tbe the ring in Example 8. Let S= {fET|2) = 0}. Prove that Sis a

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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