#23 on the picture I sent. Thomas W. Hungerford - Abstrac xO 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 moreof 621+A) Read aloudV DrawF HighlightO Erase78with 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 = MN 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 Z1g 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?Ote 2012 Cp La A Rig taed May at be opind cd ar we ar la pt Dete droie d perty cotmy be eBodtdtr Cp Bralvew tadd te ny t doat dty ha he ov rngpet Cngge Laming righeo eddo y tme i gh o i56Chapter 3 Rings22. Define a new addition O and multiplication O on Z bya Ob = a + b - 1andaOb = a + b – ab,where the operations on the right-hand side of the equal signs are ordinaryaddition, subtraction, and multiplication. Prove that, with the new operationsO and O, Z is an integral domain.23. Let E be the set of even integers with ordinary addition. Define a new sesatmultiplication on E by the rule “a * b = ab/2" (where the product on theright is ordinary multiplication). Prove that with these operations E is acommutative ring with identity.24. Define a new addition and multiplication on Z bya O b = a + b – 1anda O b = ab - (a + b) + 2.Prove that with these new operations Z is an integral domain.25. Define a new addition and multiplication on Q byrOs=r+s+1andrOs = rs + r+s.Prove that with these new operations Q is a commutative ring with identity. Isit an integral domain?

Question

#23 on the picture I sent. Thomas W. Hungerford - Abstrac xO 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 moreof 621+A) Read aloudV DrawF HighlightO Erase78with 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 = MN 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 Z1g 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?Ote 2012 Cp La A Rig taed May at be opind cd ar we ar la pt Dete droie d perty cotmy be eBodtdtr Cp Bralvew tadd te ny t doat dty ha he ov rngpet Cngge Laming righeo eddo y tme i gh o i56Chapter 3 Rings22. Define a new addition O and multiplication O on Z bya Ob = a + b - 1andaOb = a + b – ab,where the operations on the right-hand side of the equal signs are ordinaryaddition, subtraction, and multiplication. Prove that, with the new operationsO and O, Z is an integral domain.23. Let E be the set of even integers with ordinary addition. Define a new sesatmultiplication on E by the rule “a * b = ab/2&#34; (where the product on theright is ordinary multiplication). Prove that with these operations E is acommutative ring with identity.24. Define a new addition and multiplication on Z bya O b = a + b – 1anda O b = ab - (a + b) + 2.Prove that with these new operations Z is an integral domain.25. Define a new addition and multiplication on Q byrOs=r+s+1andrOs = rs + r+s.Prove that with these new operations Q is a commutative ring with identity. Isit an integral domain?

Accepted Answer

Let E be the set of even integers where addition is defined as ordinary addition and multiplication…

Let Ebe the set of even integers with ordinary addition. Define a new multiplication on E by the rule Tawbi- ab/2" (where the product on the right is ordinary multiplication). Prove that with these operations Eisa commutative ring with identity