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Rings with generalized identities. (English) Zbl 0847.16001
Pure and Applied Mathematics, Marcel Dekker. 196. New York, NY: Marcel Dekker. xi, 522 p. (1996).
The purpose of this book is to give a self-contained and rigorous account of the theory of generalized identities and some consequences of it. Because of the complexity of the material, our comments will generally be limited to simple or special cases. The notion of a generalized identity is best explained in steps. If \(R\) is an algebra over a field \(F\) and \(p(x)=p(x_1,\dots,x_n)\in F\{X\}\), the free noncommutative algebra over \(F\) in indeterminates \(X\), then \(p(x)\) is a polynomial identity for \(A\subseteq R\) if \(p(a_1,\dots,a_n)=0\) for all substitutions with \(a_i\in A\). If instead, \(p(x)\in R\coprod_FF\{X\}\), the coproduct (or free product) over \(F\) of \(R\) and \(F\{X\}\), then \(p(x)\) is a generalized polynomial identity (GPI) for \(A\subseteq R\) if again all \(p(a_1,\dots,a_n)=0\). A generalized identity is obtained by applying certain endomorphisms of \(R\) to the variables in \(p(x)\) in the following sense. Let \(W\) be all formal words (or free semi-group) in the set of derivations, automorphisms, and anti-automorphisms of \(R\), \(X^W=X\cup\{x^w_i\mid x_i\in X\) and \(w\in W\}\) be indeterminates over \(F\), and \(p(x_i^{w_j})\in R\coprod_FF\{X^W\}\). Then \(p(x^{w_j}_i)\) is a generalized identity (GI) for \(A\subseteq R\) if \(p(a^{w_j}_i)=0\) for all \(a_i\in A\), where \(a_i^{w_j}=(a_i)f_1\circ f_2\circ\cdots\circ f_k\) when \(w_j=f_1 f_2\cdots f_k\). A few simple examples using \(x\) and \(y\) for indeterminates will illustrate how such identities can arise. If \(D\) is a derivation of \(R\) then the image \(A^D\) of \(A\subseteq R\) under \(D\) is commutative when \(A\) satisfies the GI \(x^Dy^D-y^Dx^D\); the product \(DH\) of derivations \(D\) and \(H\) of \(R\) is itself a derivation exactly when \(x^Dy^H+x^Hy^D\) is a GI for \(R\); a derivation \(D\) of \(R\) is algebraic over \(F\) when \(x^{D^n}+x^{D^{n-1}}c_{n-1}+\dots+x^Dc_1\), for \(c_j\in F\), is a GI for \(R\); and if \(R\) has an involution *, then the images of the symmetric elements under the derivation \(D\) commute when \(R\) satisfies the GI \((x^D+x^{*D})(y^D+y^{*D})-(y^D+y^{*D})(x^D+x^{*D})\). Some GIs, such as \(x^{D+H}-x^D-x^H\) are trivial in the sense that they hold formally and give no information about \(R\).
The original results about rings satisfying a GI were primarily for prime rings and considered only derivations and automorphisms. The arguments were very computational and complex, and required replacing the given GI by another having special “exponents” and also “equivalent” modulo the trivial identities. It is difficult to make these notions precise. Subsequent extensions to semiprime rings using the method of orthogonal completions, and the consideration of derivations, automorphisms, and anti-automorphisms greatly increased the notational, technical, and computational complexity of the subject. In addition, much of the auxiliary material, in the form required, is not readily accessible in texts or in the literature.
The authors have done a commendable job of presenting this involved and complex subject matter in a clear, concise, and coherent manner. The text provides a brief but well written account of the necessary background material, particularly on the diamond lemma, rings of quotients, orthogonal completions, and primitive rings and becomes a single worthwile reference for these. The type is large and very readable, and the text is remarkably free of misprints. If one is interested primarily in prime rings, then much of the most difficult material can be avoided and statements of the results for semiprime rings easily yield the appropriate results for prime rings. Given the complexity of the material, a somewhat more leisurely presentation with a bit more informal discussion of the material and more examples than the very few presented in the text would have been a worthwhile addition to help the reader.
The book consists of nine chapters, the first five of which cover the necessary background material. Chapter 1 begins with tensor products and free algebras, then discusses the diamond lemma about the uniqueness of normal forms, and ends with first order formal logic leading to Horn formulas. Chapter 2 primarily concerns the maximal right (Utumi) quotient ring, the Martindale quotient ring, and the symmetric quotient ring for semiprime rings, as well as the notion of the extended centroid. Chapter 3 is about the method of orthogonal completions, which is essential for the authors’ treatment of semiprime rings. Chapter 4 centers on the structure of a primitive ring with nonzero socle, and also discusses derivations and involutions for such rings. Chapter 5 deals with (restricted) Lie algebras and introduces the notion of (restricted) differential Lie algebras, and is particularly concerned with bases for and computations in these.
The focus of the book begins with Chapter 6 which starts with Martindale’s fundamental GPI theorem for prime rings, showing that a prime ring \(R\) satisfies a GPI exactly when its central closure \(RC\) is primitive with nonzero socle and \(eRe\) is a finite dimensional algebra over \(eC\) for each \(e^2=e\in\text{soc}(RC)\). When \(R\) satisfies a GI involving only one anti-automorphism (and no derivation or automorphism), \(R\) must satisfy a GPI, and this is extended further to semiprime rings. Chapter 6 also shows that a GPI for a semiprime ring \(R\) holds for the maximal right quotient ring of \(R\), and a GI using one anti-automorphism holds for the symmetric quotient ring of \(R\). Finally, a description of the \(T\)-ideal of GPIs for prime and for semiprime rings is given. Chapter 7 deals in full generality with GIs for prime rings \(R\). The authors construct certain rings associated with \(R\) and use these to present a formal theory of GIs, trivial GIs and reduced GIs. The main results show that any prime ring \(R\) satisfying a reduced (nontrivial) GI satisfies a GPI, the reduced GI is satisfied by the symmetric quotient ring of \(R\), and the indeterminates \(x^{w_j}_i\) of the reduced GI \(p(x^{w_j}_i)\) can essentially be replaced with independent indeterminates \(y_{ij}\), so the reduced GIs arise from substitution into GPIs for \(R\). Some applications of these results to algebraic derivations and to products of derivations are given. Chapter 8 is a very technical one which extends the results in Chapter 7 to semiprime rings. The results in this case are more intricate, and use the notion of Frobenius (anti-)automorphism in an essential way. Finally, Chapter 9 presents a paper of the authors showing that with suitable additional restrictions, any Lie isomorphism between the skew-symmetric elements of two prime rings, each with involution, extends to an associative isomorphism between the subrings generated by these skew-symmetric elements.

16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16R50 Other kinds of identities (generalized polynomial, rational, involution)
16N60 Prime and semiprime associative rings
16W25 Derivations, actions of Lie algebras
16W20 Automorphisms and endomorphisms
17A30 Nonassociative algebras satisfying other identities
17B01 Identities, free Lie (super)algebras
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
16W10 Rings with involution; Lie, Jordan and other nonassociative structures