Abstract Algebra Answers

Prove the fact: Let 0 → A → B → C → 0 be an exact sequence of (additive) abelian groups. If, for some prime p, C is p-torsionfree, and p^n*A = 0 for some n, then the exact sequence splits.

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.

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

For any k-algebra R and any finite field extension K/k, show that rad R is nilpotent iff rad R^K is nilpotent.

For any ring R with (Jacobson) radical J, show that the power series ring A = R[[xi : i ∈ I]] can never be J-semisimple if R is nonzero.

For any ring R with (Jacobson) radical J, show that the power series ring A = R[[xi : i ∈ I]] (in a nonempty set of, say, commuting variables {xi}) has radical J +(sum over i in I)Axi.

Let R be any ring whose additive group is torsion-free. Show (without using Amitsur’s Theorem) that J = rad R[t] <> 0 implies that R ∩ J <> 0.

Give an example of a ring R with rad R is nonzero but rad R[t] = 0.

Given an element a in a commutative ring R, let S be the multiplicative set 1 + aR ⊆ R. Does a ∈ rad(S−1R) hold if S ⊆ R is taken to be only a multiplicative set containing 1 + aR?

Given an element a in a commutative ring R, let S be the multiplicative set 1 + aR ⊆ R.
Show that a ∈ rad(S−1R) (where a means a/1 ∈ S−1R).