Kakatiya University (KU) 2010-2nd Year B.Sc Mathematics B.A./ (ABSTRACT ALGEBRA
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b) obtain ker (f) and f (18) for f : Z ® Z10 such that f(1) = 6.
UNIT-II
19. a). obtain all units in the ring Z x Z.
b). obtain the solutions of the formula x2 + x - six = 0 in the ring Z14 by factoring the quadratic polynomial.
20. a). obtain all units in the ring Z5.
b). obtain all solutions of the formula x3 - 2x2 - 3x = 0 in Z12.
21. a). obtain all units in the ring Z x Q x Z.
b). Solve the formula 3x = two in the field Z7
22. a). obtain all units in the ring Z4.
b). obtain the Characteristic of the ring Z3 x Z4.
23. a). obtain all units in the matrix ring M2(Z2).
b). obtain the Characteristic of the ring Z x Z.
24. a). obtain all solutions of the formula x2+2x+2 = 0 in Z6.
b). obtain the characteristic of the ring Z3 x 3Z.
25. a). obtain the characteristic of the ring 2Z.
b). Let R be a commutative ring with unity of characteristic 4. calculate and simplify (a + b)4 for a, b ? R.
26. a). obtain the characteristic of the ring Z6 x Z15.
b). Let R be a commutative ring with unity of characteristic 3. calculate and simplify (a + b)6 for a, b ? R.
27. a). obtain the characteristic of the ring Z3 x Z3.
b). obtain all prime ideals and all maximal ideals of Z2 x Z2.
28. a). obtain all solutions of x2 - 2x + four = 0 in Z6.
b). obtain all prime ideals and all maximal ideals of Z6.
29. a). obtain all prime ideals and all maximal ideals of Z12.
b). obtain a prime ideal of Z x Z that is not maximal.
30. Show that for a field F , the set S of all matrices of the form for a, b ? F
is a right ideal but not a left ideal of M2(F). Is 'S' a sub ring of F?
31. Let A and B be ideals of a commutative ring R. The quotient A : B of A by B is
described by A: B { r ? R / r b ? A for all b ? B}.
Show that A: B is an ideal of R.
32. Let R and R' be 2 rings and let f : R® R' be a ring homomorphism such that f(R) ? {0' }. Show that if R has unity one and R' has no zero devisors then f(1) is unity for R'.
| Earning: Approval pending. |
