##
**Free rings and their relations.**
*(English)*
Zbl 0232.16003

London Mathematical Society Monographs. No. 2. London-New York: Academic Press. xvi, 346 p. £7.00; $ 19.50 (1971).

The object of this book is to present an account of free algebras and related rings, as far as they are known today. The basic objects of study are firs (free ideal rings, i.e. rings whose right ideals are free) and their variants. This is a class of rings, arising rather naturally and including free algebras as well as many other types. In fact it stands in the same relation to free algebras as commutative principal ideal domains do to polynomial rings in one variable over a field. Much of the theory of firs is a natural generalization of the commutative theory. Let us briefly indicate the topics covered. Some parts of the book use lattice theory and homological algebra. An appendix gives a brief summary of the relevant facts.

Since most of the material has not appeared in book form before, it was necessary to start at the beginning. A good deal of background material not readily available in the literature has been included in chapter 0. It treats for example the special form of Morita theory, the projective trivial rings, Kaplansky’s theorem on projective modules, local rings, eigenrings (the eigenring of a right ideal \(A\) is the quotient ring \(I(A)/A\), where \(I(A)\) is the idealizer of \(A)\), centralizers, the Ore construction of rings of fractions, as well as some other methods of embedding of semigroups in groups, modules over Ore rings, some unexpected consequences of Ore conditions, free associative algebras, skew polynomial rings.

Chapter 1 is devoted to the definition and basic properties of firs. If one only demands that all finitely generated right ideals, or all \(n\)-generated right ideals, shall be free of unique rank, one obtains the wider classes of semifirs and \(n\)-firs respectively. These notions are automatically left-right symmetric and arise naturally if one postulates that certain dependence relations should be trivializable. A special case, which arises when we insist on “trivializing” our relations by subgroups of the general linear group is discussed. This will be of importance when we come to meet firs (such as free associative algebras) in which every invertible matrix is a product of elementary and diagonal ones. From the homological point of view, firs are rather trivial. They are hereditary rings in which every projective is free (of unique rank). It is possible to describe not merely firs, but also matrix rings over firs in this way, and in fact it gives a categorically invariant description of these classes. This is done with a description of flat modules over semifirs. The link with weak Bezout rings (they are just the 2-firs) is established. One of the parts of this chapter is devoted to the notion of inertia, useful for studying extensions of firs.

Chapter 2 treats analogies of the Euclidean algorithm. Just as the Euclidean algorithm singles out some important classes of principal ideal domains, so there are some special classes of firs that may be described by a “weak” algorithm. Corresponding to \(n\)-firs, we have the \(n\)-term weak algorithm for any filtered ring. It turns out that \(n\)-term weak algorithm on a filtered ring can be described entirely in terms of the associated graded ring. The weak algorithm can be used to characterize free algebras. It includes Bergman’s classification of all rings with a weak algorithm. Then it turns to a closer study of the 2-term weak algorithm, using non-commutative continuants, and to a presentation of the \(\mathrm{GL}_2\) for such rings. The non-commuting power series rings can be characterized in analogous fashion by an “inverse weak algorithm”, which is in many ways simpler than the weak algorithm. However it provides some very explicit information on the relations in free power series rings. Algorithms with transfinite range are considered. This is useful for constructing one-sided counter-examples.

In chapter 3 after reviewing the familiar notion of a commutative unique factorization domain (UFD) we can find its generalization, obtained by looking at the lattice of factors. The resulting notion of non-commutative UFD is mainly of interest in the case of 2-firs. Then different aspects of factorization, including the notion of rigid UFD, which is the non-commutative analogue of a discrete valuation ring, are under consideration.

The chapter 4 examines more closely those 2-firs in which the lattice of factors of any non-zero element is distributive. It is shown that this holds for free algebras, and consequences are traced out, including the form eigenrings take in this case.

The main topic of the chapter 5 is the study of a class of modules over firs which forms a natural generalization of torsion modules over principal ideal domains (and to which they reduce in the commutative case). These “torsion” modules also obey a duality, valid in a rather more general context. The remainder of the chapter deals with the intersection theorem for firs and the interrelation of various chain conditions.

The chapter 6 studies subrings of firs and semifirs which satisfy some finiteness or commutativity conditions. The main results describe the form taken by the centre: a commutative ring can be the centre of a 2-fir if and only if it is an integrally closed integral domain, and it can be the centre of a principal ideal domain if and only if it is a Krull domain. Further, the centre of a fir that is not principal is a field. Some sections are devoted to a study of invariant elements in 2-firs and their factors and some of them examine subalgebras and centralizers in free algebras. The chapter ends with a fundamental result on free algebras: Bergman’s centralizer theorem.

By contrast chapter 7 is concerned with quite general rings. This chapter studies ways of embedding rings in fields, or more generally, the homomorphisms of rings into fields. After some generalities on the rings of fractions, obtained by inverting matrices, the notion of a matrix ideal is introduced. This corresponds to the concept of an ideal in a commutative ring, but has no direct interpretation. The analogue of a prime ideal (so called the prime matrix ideal) is defined. It has properties corresponding closely to those of prime ideals. What is more, the prime matrix ideal can be used to describe the homomorphisms of general rings into fields, just as prime ideals are used in the commutative case. This follows from the characterization of the prime matrix ideals as the sets of matrices which become singular under a homomorphism into some field. This characterization is applied to derive criteria for a general ring to be embeddable in a field, or to have a universal field of fractions.

The final Chapter 8 returns to the case of principal ideal domains. There is one important property of principal ideal domains which does not carry over to firs: the diagonal reduction of matrices. In particular results of this chapter apply to skew polynomial rings and they allow to analyse certain simple types of skew field extensions. The concluding section describes Jategaonkar’s iterated skew polynomial rings and some of their properties.

Although on the face of it, the topic of this book is rather specialized, the main aim has been to bring out the connection of two major branches of ring theory (the theory of (usually non-commutative) algebras and the theory of commutative rings) and to emphasize that all is part of the same subject. Secondly, the information on free rings provided in this book will help to further the study of “non-commutative algebraic geometry”. Thirdly, the methods used here on firs give some indication of what is needed to study more general rings.

Throughout the book a substantial number of exercises and open problems has been included.

Since most of the material has not appeared in book form before, it was necessary to start at the beginning. A good deal of background material not readily available in the literature has been included in chapter 0. It treats for example the special form of Morita theory, the projective trivial rings, Kaplansky’s theorem on projective modules, local rings, eigenrings (the eigenring of a right ideal \(A\) is the quotient ring \(I(A)/A\), where \(I(A)\) is the idealizer of \(A)\), centralizers, the Ore construction of rings of fractions, as well as some other methods of embedding of semigroups in groups, modules over Ore rings, some unexpected consequences of Ore conditions, free associative algebras, skew polynomial rings.

Chapter 1 is devoted to the definition and basic properties of firs. If one only demands that all finitely generated right ideals, or all \(n\)-generated right ideals, shall be free of unique rank, one obtains the wider classes of semifirs and \(n\)-firs respectively. These notions are automatically left-right symmetric and arise naturally if one postulates that certain dependence relations should be trivializable. A special case, which arises when we insist on “trivializing” our relations by subgroups of the general linear group is discussed. This will be of importance when we come to meet firs (such as free associative algebras) in which every invertible matrix is a product of elementary and diagonal ones. From the homological point of view, firs are rather trivial. They are hereditary rings in which every projective is free (of unique rank). It is possible to describe not merely firs, but also matrix rings over firs in this way, and in fact it gives a categorically invariant description of these classes. This is done with a description of flat modules over semifirs. The link with weak Bezout rings (they are just the 2-firs) is established. One of the parts of this chapter is devoted to the notion of inertia, useful for studying extensions of firs.

Chapter 2 treats analogies of the Euclidean algorithm. Just as the Euclidean algorithm singles out some important classes of principal ideal domains, so there are some special classes of firs that may be described by a “weak” algorithm. Corresponding to \(n\)-firs, we have the \(n\)-term weak algorithm for any filtered ring. It turns out that \(n\)-term weak algorithm on a filtered ring can be described entirely in terms of the associated graded ring. The weak algorithm can be used to characterize free algebras. It includes Bergman’s classification of all rings with a weak algorithm. Then it turns to a closer study of the 2-term weak algorithm, using non-commutative continuants, and to a presentation of the \(\mathrm{GL}_2\) for such rings. The non-commuting power series rings can be characterized in analogous fashion by an “inverse weak algorithm”, which is in many ways simpler than the weak algorithm. However it provides some very explicit information on the relations in free power series rings. Algorithms with transfinite range are considered. This is useful for constructing one-sided counter-examples.

In chapter 3 after reviewing the familiar notion of a commutative unique factorization domain (UFD) we can find its generalization, obtained by looking at the lattice of factors. The resulting notion of non-commutative UFD is mainly of interest in the case of 2-firs. Then different aspects of factorization, including the notion of rigid UFD, which is the non-commutative analogue of a discrete valuation ring, are under consideration.

The chapter 4 examines more closely those 2-firs in which the lattice of factors of any non-zero element is distributive. It is shown that this holds for free algebras, and consequences are traced out, including the form eigenrings take in this case.

The main topic of the chapter 5 is the study of a class of modules over firs which forms a natural generalization of torsion modules over principal ideal domains (and to which they reduce in the commutative case). These “torsion” modules also obey a duality, valid in a rather more general context. The remainder of the chapter deals with the intersection theorem for firs and the interrelation of various chain conditions.

The chapter 6 studies subrings of firs and semifirs which satisfy some finiteness or commutativity conditions. The main results describe the form taken by the centre: a commutative ring can be the centre of a 2-fir if and only if it is an integrally closed integral domain, and it can be the centre of a principal ideal domain if and only if it is a Krull domain. Further, the centre of a fir that is not principal is a field. Some sections are devoted to a study of invariant elements in 2-firs and their factors and some of them examine subalgebras and centralizers in free algebras. The chapter ends with a fundamental result on free algebras: Bergman’s centralizer theorem.

By contrast chapter 7 is concerned with quite general rings. This chapter studies ways of embedding rings in fields, or more generally, the homomorphisms of rings into fields. After some generalities on the rings of fractions, obtained by inverting matrices, the notion of a matrix ideal is introduced. This corresponds to the concept of an ideal in a commutative ring, but has no direct interpretation. The analogue of a prime ideal (so called the prime matrix ideal) is defined. It has properties corresponding closely to those of prime ideals. What is more, the prime matrix ideal can be used to describe the homomorphisms of general rings into fields, just as prime ideals are used in the commutative case. This follows from the characterization of the prime matrix ideals as the sets of matrices which become singular under a homomorphism into some field. This characterization is applied to derive criteria for a general ring to be embeddable in a field, or to have a universal field of fractions.

The final Chapter 8 returns to the case of principal ideal domains. There is one important property of principal ideal domains which does not carry over to firs: the diagonal reduction of matrices. In particular results of this chapter apply to skew polynomial rings and they allow to analyse certain simple types of skew field extensions. The concluding section describes Jategaonkar’s iterated skew polynomial rings and some of their properties.

Although on the face of it, the topic of this book is rather specialized, the main aim has been to bring out the connection of two major branches of ring theory (the theory of (usually non-commutative) algebras and the theory of commutative rings) and to emphasize that all is part of the same subject. Secondly, the information on free rings provided in this book will help to further the study of “non-commutative algebraic geometry”. Thirdly, the methods used here on firs give some indication of what is needed to study more general rings.

Throughout the book a substantial number of exercises and open problems has been included.

### MSC:

16E60 | Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |

16S50 | Endomorphism rings; matrix rings |

16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |

16S36 | Ordinary and skew polynomial rings and semigroup rings |

16U20 | Ore rings, multiplicative sets, Ore localization |

16B50 | Category-theoretic methods and results in associative algebras (except as in 16D90) |

16U30 | Divisibility, noncommutative UFDs |