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?Solution: Chebotar, Lee and Puczylowski. 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. 1992 (Lanski): 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?1993 (Puczylowski): Suppose that R is a nil ring. Is then R{X} Brown-McCoy radical if card X>1?Solution: Chebotar, Ke, Lee and Puczylowski. 1993 (Puczylowski): Suppose Solution: Chebotar, Ke, Lee and Puczylowski.A and B are nil algebras. Is their tensor product algebra Brown-McCoy radical?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, butR[X,X^{-1};f,d] does?Solution: Chebotar. |