Open Problems Solved

The problems are ordered by their publication dates.

1961 (Herstein):
1. Problems 2 and 3 (the description of Lie mappings and Lie derivations for simple rings - MC) in the setting of the simple rings [R, R]  are worthy of serious attention.
Solution: Beidar and Chebotar (Lie homomorphisms) and (Lie derivations).
2. To settle the very difficult questions above for the skew-symmetric elements, K, of a single ring and their associated simple Lie rings [K, K] offers a real challenge. 
Solution: Beidar, Bresar, Chebotar and Martindale (Lie homomorphisms of  [K, K])  and (Lie derivations of [K, K]). 
3. If R is a simple ring with a nonzero zero-divisor and if T is a subring of R invariant under all the automorphisms of R, is T a subset of Z or T=R?
4. If R is a simple ring, and if the niipotent elements R form a subring W of R, is W=(0) or W=R?
5. If R is a simple ring with a nontrivial zero-divisor, is every ab — ba in R a sum of nilpotent elements?

In the preceding example (their solution of the problem by Goodearl and Warfield - MC), every annihilator ideal is nilpotent. In fact, we do not know of an L-prime ring which does not have this property.
Two situations for which we do not have examples are when a product of both inner and outer derivations is a nonzero derivation in characteristic zero, or when p or fewer such gives a nonzero derivation in characteristic p. Perhaps such products can be derivations only if the product is zero.
1993 (Bresar and Vukman):
We conclude with an open question: is it possible to  generalize the Theorem by proving that U (the subring of a prime ring generated by [D(x),x)], where D is a nonzero derivation - MC) contains  a nonzero  two-sided  ideal?
The reader will no doubt have noticed that we left the following questions unanswered.
Question 1. Is there an n-transitive Jordan algebra A with a nonzero Jordan ideal J which is not n-transitive?
Question 2. Is there an n-transitive Jordan  algebra which is not (n + 1)-transitive for any n > 1?
Suppose that R is a nil ring. Is then R{X} Brown-McCoy radical if card X>1?
Solution: Chebotar, Ke, Lee and Puczylowski.

Suppose A and B are nil algebras. Is their tensor product algebra Brown-McCoy radical?
Solution: Chebotar, Ke, Lee and Puczylowski.

Suppose that R is a nil ring. Is the ring of polynomials in two or more commuting indeterminates over R Brown-McCoy radical?
Solution: Chebotar, Ke, Lee and Puczylowski.

1999 (Bresar and Semrl):
We leave as an open question whether this (the main result of the section, Theorem 4.1, without assumption that derivations commute - MC) remains true in more general algebras (possibly in dense algebras of linear operators).
2005 (Grunenfelder, Omladic and Radjavi):
We do not know whether in odd dimensions transitivity always implies 2-transitivity. No counter examples are known to us at this time.
2007 (Beidar):
1. Does there exist a prime ring with zero center whose central closure is a simple ring with an identity element?
Solution:Chebotar.
2. Does there exist a prime nil ring whose central closure is a simple ring with identity?
Solution: Chebotar, Ke, Lee and Puczylowski.

1. Is there a locally nilpotent ring R and a derivation d such that R[X:d] maps onto a ring with a non-zero idempotent?
Solution: Chebotar.
2. Suppose R is a locally nilpotent ring with an automorphism f and an f-derivation d. Is it possible that R admits no nonzero idempotents, but
R[X,X^{-1};f,d] does?
Solution: Chebotar.