This paper makes critical use of a new methodology, namely, the theory of "functional identities" with the key notion of "d-freeness." This new area of ring theory was invented and developed by my colleagues Bresar, Beidar, and Chebotar and has already proved useful in solving other problems. The four of us are currently in the process of writing a treatise entitled "Functional Identities."

  Could you summarize the significance of your paper in layman's terms?

This is very difficult, if not impossible, to do, since the area occupies a corner of abstract algebra and thereby would probably only make sense to someone who has had a first-year graduate course in modern algebra. However, I'll make a try. An "associative ring" is a set of elements with two binary operations, called "addition" and "multiplication," which satisfies several basic and natural axioms (e.g. both addition and multiplication obey the associative law, but multiplication is not assumed to be commutative). Familiar examples are the integers, rational numbers, real numbers, and the complex numbers. A more exotic example would be a "Boolean" ring, in which every element is idempotent, i.e. equal to its own square. From the standpoint of "noncommutative" ring theory the prototype example is the set of all linear transformations of a (possibly infinite dimensional) vector space. The basic "building blocks" of noncommutative ring theory are the so-called "prime" rings (of which the preceding example is one of many). Two rings, R and S, are said to be "isomorphic" (i.e., essentially equal) if there is a one-to-one map f of R onto S such that f(x + y) = f(x) + f(y) and f(xy) = f(x)f(y), i.e., f preserves the operations. An associative ring R may be turned into a "Lie ring" by keeping the same addition but defining a new multiplication: [x,y] = xy - yx. Now we come to the questions posed by Herstein in 1961: (A) if two prime rings R and S are isomorphic as Lie rings must they be isomorphic as associative rings (or at least close to being isomorphic)? (B) (more difficult) if the "skew" elements of two prime rings with "involution" are isomorphic as Lie rings, must R and S be close to being isomorphic as associative rings? Under the additional assumption of the presence of some idempotent elements I (with my students Swain and Blau) essentially answered these questions. A major breakthrough was made by Bresar (1993) when he solved (A) without the idempotent assumption. Beidar, Mikhalev, and I (1994) solved one aspect of (B). Then the aforementioned theory of functional identities was employed by Beidar, Bresar, Chebotar, and I to completely solve Herstein's problems in the trilogy of papers mentioned above.

  How did you become involved in this research?

Herstein had originally posed the Lie isomorphism problem to me as a possible thesis topic in 1956. Although I was not able to make any headway at that time, the problem continued to gnaw at me in the ensuing years and (as indicated above) I was able to solve it under the "idempotent" condition. During a visit to Moscow in 1991 (in the course of writing a book with Beidar and Mikhalev) Beidar suggested giving Herstein's problem (B) a try in view of Bresar's recent breakthrough on problem (A). With the advent of the Bresar-Beidar-Chebotar theory of functional identities it became clear that the Herstein problems could then be completely solved.