Let R be a left Noetherian ring. For any ideal I of R let \({\mathcal C}(I)\) denote the set of elements c of R such that whenever \(r\in R\) and cr\(\in I\) or rc\(\in I\) then \(r\in I\). Given an ideal I, the ring R satisfies the left Ore condition with respect to \({\mathcal C}(I)\) provided for all \(r\in R\), \(c\in {\mathcal C}(I)\) there exist r’\(\in R\), \(c\in {\mathcal C}(I)\) such that \(c'r=r'c\). About 25 years ago, A. W. Goldie observed that if P is a prime ideal of R such that R satisfies the left Ore condition with respect to \({\mathcal C}(P)\) then \(K=\{r\in R:\) \(cr=0\) for some \(c\in {\mathcal C}(P)\}\) is an ideal of R, \({\mathcal C}(P)\subseteq {\mathcal C}(K)\), \(\bar R=R/K\) satisfies the left Ore condition with respect to \(\bar {\mathcal C}=\{c+K:\) \(c\in {\mathcal C}(P)\}\) and the classical quotient ring \(R_ P\) of the ring \(\bar R\) with respect to the set \(\bar {\mathcal C}\) has properties analogous to those of commutative local rings. This construction opened up the possibility of extending the techniques, and hence the results, of commutative algebra to a non-commutative setting, and where it worked (e.g. for enveloping algebras of finite dimensional nilpotent Lie algebras or group algebras of finitely generated nilpotent groups) this was indeed the case. Unfortunately there are many situations where it does not work. For example, let F be any field and T the ring of \(2\times 2\) upper triangular matrices with entries in F. Let \(P=\left[ \begin{matrix} 0\quad F\\ 0\quad F\end{matrix} \right]\) and \(Q=\left[ \begin{matrix} F\quad F\\ 0\quad 0\end{matrix} \right]\). Then T is a left and right Artinian ring, P and Q are the only prime ideals of T and T satisfies the left Ore condition with respect to \({\mathcal C}(Q)\) but not with respect to \({\mathcal C}(P)\). However if \(N=P\cap Q\) then \({\mathcal C}(N)\) consists of all the units of T and hence T satisfies the left Ore condition with respect to \({\mathcal C}(N)\). This example among others led the author and others to focus attention on semiprime rather than prime ideals.
Let S be a semiprime ideal of R. Call S left localizable provided R satisfies the left Ore condition with respect to \({\mathcal C}(S)\); in this case if \(K=\{r\in R:\) \(cr=0\) for some \(c\in {\mathcal C}(S)\}\) and \(R_ S\) is the ring obtained from S in the same way \(R_ P\) was obtained from P then \(R_ S\) is a left Noetherian ring with Jacobson radical \(J(R_ S)=R_ SS=\{(c+K)^{-1}(r+K):\) \(r\in S\), \(c\in {\mathcal C}(S)\}\) and \(R_ S/J(R_ S)\) is a semiprime Artinian ring. A left localizable semiprime ideal S such that \(J(R_ S)\) has the left AR (i.e. Artin-Rees) property is called classically left localizable. Not every semiprime ideal is left localizable. For example if X is an idempotent semiprime ideal of a prime left Noetherian ring T then X is not left localizable. In addition not every left localizable semiprime ideal is classically left localizable.
It turns out that a semiprime ideal S of the ring R being left localizable or classically left localizable can be characterized in terms of properties of the injective hull E(R/S) of the left R-module R/S; a sample result is that S is (classically) left localizable if and only if E/A can be embedded in a direct product (sum) of copies of E, where \(E=E(R/S)\) and \(A=\{e\in E:\) \(Se=0\}\). In this context the author calls A the first layer of E. Uniform left R-modules U and V are called similar provided E(U)\(\cong E(V)\). Again with \(E=E(R/S)\) and A the first layer of E, the second layer of E is defined to be the collection of similarity classes of uniform submodules of E/A. The ideal S is left localizable if and only if U is a torsion-free (R/S)-module for all uniform submodules of E/A and in this way localization is linked to the second layer. Let U be a uniform left R-module, P the assassinator of U and \(W=\{u\in U:\) \(Pu=0\}\). The module U is called tame provided W is a torsion-free left (R/P)-module and otherwise W is a torsion left (R/P)-module and U is called wild. The ideal S is classically left localizable if and only if every uniform submodule of E/A is tame, where E, A are as above.
To investigate the second layer two main tools are employed, namely links between prime ideals and Noetherian bimodules. Given rings R’, R, a left R-, right R’-bimodule B is called a Noetherian bimodule provided R is a left Noetherian ring, R’ is a right Noetherian ring and B is a Noetherian left R-module and a Noetherian right R’-module; moreover B is called a bond if in addition \(B\neq 0\), R’ and R are both prime and B is torsion- free both as a left R-module and as a right R’-module. The existence of a bond has considerable implications for the rings R’ and R, for example R’ is simple Artinian if and only if R is simple Artinian, and several other properties are shown to be shared by R’ and R.
Suppose from now on that R is left and right Noetherian. Let P and Q be prime ideals of R. An ideal A of R is called a right link from Q to P provided \(QP\subseteq A\subset Q\cap P\) and \(_{(R/Q)}((Q\cap P)/A)_{(R/P)}\) is a bond; in this case Q is said to be linked to P and this is written \(Q\rightsquigarrow P\) via A or simply \(Q\rightsquigarrow P\). The clique of P consists of P together with all prime ideals Q for which there exist a positive integer n and prime ideals \(P_ i\) (1\(\leq i\leq n)\) such that \(Q\rightsquigarrow P_ 1\), \(P_ 1\rightsquigarrow P_ 2,...,P_{n-1}\rightsquigarrow P_ n\), \(P_ n\rightsquigarrow P\). Any clique is a countable set. The link graph of R is the directed graph whose vertices are the prime ideals of R and given prime ideals Q, P there is a directed edge \(Q\to P\) if and only if \(Q\rightsquigarrow P\). The link graph of FBN rings and of Noetherian rings of bounded uniform dimension (i.e. there is a bound to the uniform dimensions of all prime homomorphic images) are locally finite but this is not true in general.
Let \({\mathcal X}\) be a non-empty collection of prime ideals of R. Define \({\mathcal C}({\mathcal X})=\cap \{{\mathcal C}(P):\) \(P\in {\mathcal X}\}\). Call \({\mathcal X}\) classically left localizable provided R satisfies the left Ore condition with respect to \({\mathcal C}({\mathcal X})\) and in addition if \(R_{{\mathcal X}}\) is the resulting quotient ring then (i) \(R_{{\mathcal X}}/R_{{\mathcal X}}P\) is a simple Artinian ring for each \(P\in {\mathcal X}\), (ii) every left primitive ideal of \(R_{{\mathcal X}}\) has the form \(R_{{\mathcal X}}P\) for some \(P\in {\mathcal X}\), and (iii) the injective hull of every simple left \(R_{{\mathcal X}}\)-module is the union of its socle series. A prime ideal P of R satisfies the left second layer condition if every uniform module associated with the second layer of the left R- module E(R/P) is tame and \({\mathcal X}\) satisfies the left second layer condition provided every prime ideal P in \({\mathcal X}\) does. For example the minimal prime ideals of any Noetherian ring satisfy the left second layer condition. The ring R is said to satisfy the left second layer condition provided every prime ideal of R does. Since FBN-rings have no wild modules it follows that FBN-rings satisfy the second layer condition. Moreover if J is a commutative Noetherian ring and G a polycyclic-by-finite group then the group ring J[G] satisfies the second layer condition. On the other hand if \({\mathfrak g}\) is a finite dimensional Lie algebra over an algebraically closed field of characteristic 0 then its enveloping algebra U(\({\mathfrak g})\) satisfies the second layer condition if and only if \({\mathfrak g}\) is solvable, and in this case \({\mathfrak g}\) is nilpotent if and only if all cliques in U(\({\mathfrak g})\) are finite (in which case they are all trivial). Note that if R satisfies the left second layer condition then \(\cap^{\infty}_{n=1}J^ n=0\), where J is the Jacobson radical of R. A non-empty subset \({\mathcal X}\) of prime ideals of R is classically left localizable if and only if (i) \({\mathcal X}\) satisfies the left second layer condition, (ii) \({\mathcal X}\) is a union of cliques (stability condition), (iii) \(L\cap {\mathcal C}({\mathcal X})\neq \emptyset\) for any left ideal L of R such that \(L\cap {\mathcal C}(P)\neq \emptyset\) for all \(P\in {\mathcal X}\) (intersection condition), and (iv) \(P\not\subset Q\) for all P,Q\(\in {\mathcal X}\) (incomparability condition). In particular if R is a Noetherian algebra over an uncountable field and \({\mathcal X}^ a \)non-empty collection of prime ideals of R such that there is a bound on the uniform dimensions of the rings R/P (P\(\in {\mathcal X})\) then \({\mathcal X}\) satisfies the intersection condition; if in addition \({\mathcal X}\) is a clique and satisfies the left second layer condition then \({\mathcal X}\) is classically left localizable. Moreover a semiprime ideal S of a Noetherian ring R is classically left localizable if and only if the collection \({\mathcal X}\) of associated prime ideals of S satisfies the left second layer and stability conditions.
There are nine chapters. The first three are introductory covering Goldie’s Theorem and localizable semiprime ideals, the next four give an exposition of localization and its relationship to the second layer and the last two examine rings with the second layer condition and their injective modules. The chapter headings are as follows: 1. Ore’s method of localization, 2. Orders in semi-simple rings, 3. Localization at semi- prime ideals, 4. Localization, primary decomposition, and the second layer, 5. links, bonds and Noetherian bimodules, 6. The second layer, 7. Classical localization, 8. The second layer condition, 9. Indecomposable injectives and the second layer condition. In addition there is an appendix entitled ”Important classes of Noetherian rings”.
The treatment is very thorough and comprehensive and great care has been taken throughout. At every stage the reader is shown where the discussion is heading and why such a direction is reasonable. There is an extensive bibliography and numerous examples are given in the text to illustrate the theory and the problems that the theory has had to overcome. In short the book is really required reading for anyone interested in non- commutative Noetherian rings and is highly recommended.