P. M. Cohn

On the Free Product of Associative Rings.

The object of this note is twofold. In the first place we correct a mistake in [4], where it was erroneously asserted (Lemma 4.1) that the free product of semifirs R, over a semifir K (under certain conditions) is again a semifir. In fact the conditions were not enough to ensure this. However, the result holds when K is a (skew) field, and this is all that was needed in the applications made in [4]. In Section 3 below we prove a quite general result (Theoren 3.1) on dependence in a free product of integral domains over a field, from which a correct version of Lemma 4.1 of [4] follows without difficulty. At the same time some generalizations to n-firs are obtained. A second object is to improve the results obtained in [2] on zero-divisors in free products of integral domains. It turns out that the results of Section 2 of [2] can be stated more generally (and more simply) under rather weaker hypotheses (Section 2 below). In the language of [Z] the results on integral domains obtained here refer to l-firs, and it may be that they will be of use in looking for a correct generalization of Theorem 3.1 to the case where the base ring is a semifir.

Bezout rings and their subrings

The importance of principal ideal domains (PIDs), both in algebra itself and elsewhere in mathematics is undisputed. By contrast, Bezout rings, ‡ although they represent a natural generalization of PIDs, play a much smaller role and are far less well known. It is true that many of the properties of PIDs are shared by Bezout rings, but the practical value of this observation is questioned by many on the grounds that most of the Bezout rings occuring naturally are in fact PIDs. However, there are several fairly natural methods of constructing Bezout rings from other rings, leading to wide classes of Bezout rings which are not PIDs, and it is the object of this paper to discuss some of these methods.

Some remarks on the invariant basis property

The conditions I-III occur frequently among the hypotheses in theorems about rings, both in algebra and topology (for examples of the latter, see [l] and the references given there). In particular, I is known as the invariant basis property or invariant basis number (IBN). Each of I-III fails to hold only for what may be regarded as pathological rings, but it is not at all easy to decide whether a given ring has any one of these properties. In these circumstances it is reasonable to confine oneself to

Further Algebra and Applications

1. Universal algebra.- 2. Homological algebra.- 3. Further group theory.- 4. Algebras.- 5. Central simple algebras.- 6. Representation theory of finite groups.- 7. Noetherian rings and polynomial identities.- 8. Rings without finiteness assumptions.- 9. Skew fields.- 10. Coding theory.- 11. Languages and automata.- List of Notations.- Author Index.

On a Generalization of the Euclidean Algorithm

In any valuated ring R , more specifically, a ring with a degree-function d ( a ) as defined in §2, the notion of right R -dependence may be denned as follows: (i) The elements a 1 , …, a r of R are right R-dependent if there exist b 1 , …, b r ∈ R such that . (ii) An element a of R is said to be right R-dependent on the elements a 1 , …, a r of R if a = 0 or if there exist c 1 , …, c r ∈ R such that .

Rings with a weak algorithm

determinate over a field F. All these ideas can be generalized in a straightforward manner to the noncommutative case: the principal ideal domains again form a well-behaved though rather narrow class (cf. Jacobson [9, Chapter 3]), and the valuated rings with a Euclidean algorithm are just the skew polynomial rings k[x; S, D] over a skew field k with an automorphism S and an S-derivation D (2). One obtains a slightly larger class by taking, instead of principal ideal domains, Bezout rings, i.e., integral domains in which any finitely generated (left or right) ideal is principal but this probably amounts to not much more than allowing locally principal ideal domains. A significantly wider class of rings is obtained by taking all integral domains in which any two principal right ideals with a nonzero intersection have a sum and intersection which are again principal. These are the weak Bezout rings introduced in [6], where it is shown that a weak Bezout ring in which prime factorizations exist, is a unique factorization domain, and other decomposition theorems hold (corresponding to the primary decomposition of an ideal in a Noetherian ring). Further it is shown there that the weak Bezout rings include free associative algebras in any number of free generators over a field. It is possible to weaken the definition of the Euclidean algorithm in a similar way so as to obtain rings with a weak algorithm (cf. ?2 for the definition). This was first introduced in [4] where it was applied to prove (in effect) that in any ring R with a weak algorithm, all right ideals were free R-modules. We now continue the study of rings with a weak algorithm and in particular show that they are weak Bezout rings, so that the results of [6] become applicable (?4). This

Algebraic numbers and algebraic functions

Part 1 Fields with valuations: absolute values the topology defined by an absolute value complete fields valuations, valuation rings and places the representation by power series ordered groups general valuations. Part 2 Extensions: generalities on extensions extensions of complete fields extensions of incomplete fields Dedekind domains and the string approximation theorem extensions of Dedekind domains different and discriminant. Part 3 Global fields: algebraic number fields the product formula the unit theorem the class number. Part 4 Function fields: divisors on a function field principal divisors and the divisor class group Riemanns theorem and the speciality index the genus derivations and differentials the Riemann-Roch theorem and its consequences elliptic function fields Abelian integrals and the Abel-Jacobi theorem. Part 5 Algebraic function fields in two variables: valuations on function fields of two variables.

Factorization in Non-Commutative Power Series Rings

The study of the divisibility of polynomials in a single variable over a field is essentially based on the division algorithm, which may be stated as follows: (A) Given polynomials a, b such that there exist polynomials q and a 1 , such that† .

GENERALIZED RATIONAL IDENTITIES

Publisher Summary This chapter discusses the generalized rational identities. It presents a simple proof of the Amitsur–Bergman result with a somewhat different condition on the centers (which are still assumed infinite). The main tool is the notion of a universal skew field of fractions. This was constructed by Amitsur in the special case of free algebras, using precisely his results on rational identities. The explicit construction for universal skew fields of fractions to prove the result on rational identities is used. If R is any ring, then by afield of fractions of R, one understands a field K with an embedding R → K such that K is the field generated by the image. In the commutative case, such a K exists if R is an integral domain, and it is then unique. In general no necessary and sufficient conditions are known for a field of fractions to exist and even when it does exist, it need not be unique.

Rings and Algebras

Publisher Summary This chapter focuses on rings and algebras. The development of ring theory has been in two directions. On one hand, such instances as the quaternions and matrix algebras gave rise to the subject of linear associative algebras, which developed into the study of Artinian and Noetherian rings. On the other hand, there are the rings of algebraic integers; they are essentially Dedekind rings, which arose also in algebraic geometry as coordinate rings of curves and which formed the beginning of a very active field of study, called the commutative ring theory, which in the last few decades has had a powerful impetus from algebraic geometry. A central part of ring theory is the classification of semi-simple rings, particularly the Wedderburn theorems. There are two main approaches. The first is via semi-simple modules, possibly using homological algebra. The other is via the Jacobson structure theory, which allows the result to be derived with a minimum of assumptions.