addition and multiplication

(1) Find all the solutions of x2 – 5x + 6 = 0) in (a) Z7. (b) Z8. (c) Z4 x 75. (2) Consider Z3 with the addition and multiplication mod 3 as usual. Let R=Z3 X Z3. Define (a,b) + (a’,b’) = (a +a’, 6+b) and (a,b).(a’,6′) = (aa’ – bb’, ab’ + a’b). (a) Show that (R, +,.) is a commutative ring. (b) Show that (1,0) is the identity element for the multiplication. (C) Show that the equation x2 = -1 has exactly two solutions in R. (3) Find characteristic of the rings Z5 Z6 and Z6 x Z8 x Zg (additions and multiplication are the usual addition and multiplication). (4) Let R be ring without zero divisor. Let a, b e R and assume a # 0. Show that the equation ax = 6 can have at most one solution. Give as example where no solution exist. (5) Let R be a ring with unity. Assume a E R is a unit, show that a is not a zero divisor. (6) Exercises 18 page 174: 2, 4, 10, 12, 22, 28, 37. (7) Exercises 19: 2, 10, 12, 14.
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