Selected papers of S. A. Amitsur with commentary. Part 2. Edited by Avinoam Mann, Amitai Regev, Louis Rowen, David J. Saltman and Lance W. Small (Q2765137)

This is a joint review for parts I and II (see also Zbl 0993.01014). NEWLINENEWLINENEWLINEThese two volumes include practically all of Amitsur's papers; only a handful written in Hebrew and some expository papers are omitted. They are divided into four parts, representing the different parts of algebra covered; however, there is much interaction between the different parts and in some ways it would have been better to have the papers arranged by date of publication. A very useful addition in the commentary prefacing each part, giving an overview of the contents. NEWLINENEWLINENEWLINEI. General ring theory, introduced by Louis Rowen, has 21 papers. In three papers Amitsur laid the foundations for a general theory of radicals (independently of Kurosh). Other papers deal with Morita theory and there are some isolated papers on derivations, differential operators and categorical aspects. PI-rings form the subject of Part II, but I includes Amitsur's GPI-theorem: the introduction of generalized polynomial identities, where the variables fail to commute not only with each other but also the coefficients, was a significant breakthrough, which changed our perception of PI-theory. NEWLINENEWLINENEWLINEII. Rings satisfying a polynomial identity, introduced by Lance Small. This was one of Amitsur's first and lasting interests; following Kaplansky's pioneering paper of 1948, Amitsur proves that every semiprime PI-ring can be embedded in a full matrix ring over a commutative reduced ring. Nil-semigroups and -rings, representation theory and Hilbert basis theorem are dealt with (though strangely the GPI-paper is in Part I), as well as other aspects such as the relation to Azumaya algebras, Gelfand-Kirillov dimension and PI-rings with involution in a total of 26 papers. NEWLINENEWLINENEWLINEIII. Combinatorial polynomial identity theory, introduced by Amitai Regev, has nine papers in all, dealing with polynomial identities involving combinatorial methods, such as the Amitsur-Levitzki theorem on the identity on matrix rings. Later papers describe the various properties and interesting applications of Capelli polynomials. NEWLINENEWLINENEWLINEFinally, IV. Division algebras, introduced by David Saltman, contains 28 papers, stretching over almost 50 years, covering many aspects of division algebras, such as generic splitting fields, noncrossed products (the first construction), arithmetic functions in division algebras as well as many particular results, such as the complete classification of finite subgroups of division rings. NEWLINENEWLINENEWLINESeeing these volumes makes one realize anew what a wide field was covered by Amitsur's writings and how almost each paper had a significant impact on the development of the subject. The introductions provide a useful overview and there is a brief account of his life by Avinoam Mann. As collected works go, these volumes are a collection of short and very readable papers, which many working algebraists will want to own; they should certainly form part of every mathematics library.