Abstract Algebra Answers

char(k) = 3, and let G = S3. Compute the Jacobson radical J = rad(kG), and the factor ring kG/J.

Let k be a commutative ring and G be any group. If kG is left artinian, show that kG is right artinian.

Let k be a commutative ring and G be any group. If kG is left noetherian, show that kG is right noetherian.

Let A be a normal elementary p-subgroup of a finite group G such that the index of the centralizer CG(A) is prime to p. Show that for any normal subgroup B of G lying in A, there exists another normal subgroup C of G lying in A such that A = B × C.

Let R be a commutative domain that is not a field. Show that not always R is not J-semisimple implies R is semilocal, if R is a noetherian domain.

Let R be a commutative domain that is not a field. Show that if R is not J-semisimple then R is semilocal, if R is a 1-dimensional noetherian domain.

Let R be a commutative domain that is not a field. If R is semilocal, show that R is not J-semisimple.

For any ring R with Jacobson radical J, we have an exact sequence of groups
(∗) 1→ 1 + J → U(R) → U(R) → 1 (where R = R/J),
induced by the projection map π : R → R. Show that this sequence splits if R is a commutative rad-nil Q-algebra, or R is a commutative artinian ring.

If J is a nil ideal and the ring R/J has prime characteristic p, show that 1 + J is a p-group.

Let J be an ideal in any ring R. Suppose J^n+1 = 0. If, moreover, the group J/J2 has a finite exponent m, show that (1 + J)^m^n = 1.