Answer Happy

higher algebra
1. (i) Let R be the ring of integers. Show that M = {(a,b,c) € 23:a + 2b = 0) is a submodule of the left R-module Z3 . [4 Marks) (ii) Let g: Z → Z2 be the function given by g(x, y, z) = (2x + 4y, 3x + z). Prove that g is a morphism of left Z-modules, and give an explicit description of img 16 Marks Let S3 be the symmetric group on {1,2,3} and let T, 0 be the cycles T = (23), o = (132). In the group ring Z,S3, calculate (27 +30T). (o2 +57), expressing your answer as a linear combination of T, O, OT, TO, O2 and e. [6 Marks] (i) Let R= Z, let M be the left R-module Z, and let X be the subset, X = {10,9,8} CM. (a) Show that X generates M. [2 Marks (b) Is there any proper subset of X which also generates M? Either give an example (with justification), or explain why there is no such subset. [3 Marks) (ii) Let R = Zg, and let N = R3. Determine whether or not X = {(1,1, 2), (2,3,3), (1,7,0)} is a free set in N. [4 Marks) (iii) Let R= M2(Z), and let P be the right R-module P=Z? with the usual addition and scalar multiplication: (a,b) + (c,d) = (a + c,b+d), (a,b) (am + bq, an + br). 9 (a) Determine whether or not P is free as a right R-module. [4 Marks Let 0: R2 + P be the right R-module morphism (b) C *((" "). (* ) H (m + q + 2w – y,n+r+ 2x - 2 - ) 9 Y 2 (You do not need to prove that this is an R-linear map). Give a generating set for im o. (3 Marks