Abstract Algebra Answers

List left and right cosets of H= {epsilon, alpha} in S3, where epsilon is the identity permutation, alpha is the transposition (2,3). Is H a normal subgroup of S3? Justify.

Recursive sets are not closed under
a) complementation
b) intersection
c) substitution
d) min

Of three cards, one is painted red on both sides; one is painted black on both sides; and one is
painted red on one side and black on the other. A card is randomly chosen and placed on a table. If
the side facing up is red, what is the probability that the other-side is also red?

Prove that in any group G, orders of elements ab and ba are equal.

Let f:G1-->G2 be a homomorphism of groups. Let H be a cyclic subgroup of G1. Prove the f(H) is a cyclic subgroup of G2.

Let Ai (i ∈ I) be ideals in a ring R, and let A =(intersection on i) Ai. True or False: “If each R/Ai is von Neumann regular, then so is R/A”?

Compute the radical of the ring Tn(k) of n × n upper triangular matrices over any ring k.

For a triangular ring T =
R M
0 S
(where M is an (R, S) - bimodule), show that rad(T) =
rad(R) M
0 rad(S)

Show that, for any direct product of rings Ri, rad ((direct product)Ri) = (direct product) rad Ri.

Let Ai (i ∈ I) be ideals in a ring R, and let A =(intersection on i) Ai. True or False: “If each R/Ai is J-semisimple, then so is R/A”?