Universal Field Of Fractions Research Articles

A standard form in (some) free fields: How to construct minimal linear representations

AbstractWe describe a standard form for the elements in the universal field of fractions of free associative algebras (over acommutativefield). It is a special version of the normal form provided by Cohn and Reutenauer and enables the use of linear algebra techniques for the construction of minimal linear representations (in standard form) for the sum and the product of two elements (given in a standard form). This completes “minimal” arithmetic in free fields since “minimal” constructions for the inverse are already known. The applications are wide: linear algebra (over the free field), rational identities, computing the left gcd of two non-commutative polynomials, etc.

A FACTORIZATION THEORY FOR SOME FREE FIELDS

Although in general there is no meaningful concept of factorization in fields, that in free associative algebras (over a commutative field) can be extended to their respective free field (universal field of fractions) on the level of minimal linear representations . We establish a factorization theory by providing an alternative definition of left (and right) divisibility based on the rank of an element and show that it coincides with the " classical'' left (and right) divisibility for non-commutative polynomials. Additionally we present an approach to factorize elements, in particular rational formal power series, into their (generalized) atoms. The problem is reduced to solving a system of polynomial equations with commuting unknowns.

The Universal Field of Fractions of a Semifir: III. Centralizers and Normalizers

The Universal Field of Fractions of a Semifir: III. Centralizers and Normalizers

On the abelianized multiplicative group of a universal field of fractions

On the abelianized multiplicative group of a universal field of fractions

On the law of nullity

In chapter 7 of (2) conditions were given for a ring to be embeddable in a skew field; in particular, it was shown that any semifir has a universal field of fractions, over which all full matrices can be inverted. This was generalized in two different directions, by Bergman (in a letter to one of the authors in 1971) and by Dicks and Sontag(7). Dicks and Sontag characterized those rings having a field of fractions in which all full matrices are inverted; they showed that this is equivalent to Sylvester's law of nullity, and further showed that this forces the ring to have weak global dimension not exceeding 2 and all finitely generated projective modules to be free. Bergman on the other hand investigated weakly semihereditary rings having a rank function on projective modules which takes values in the natural numbers. He showed that there was a homomorphism from any such ring to a field of fractions in which every full map between finitely generated projective modules is inverted. Weakly semihereditary rings with a rank function to the natural numbers are the analogue of semifirs and so it is natural to look for a characterization of rings with a rank function on projective modules such that all full maps between projective modules become invertible in a suitable field of fractions. We shall find that, as before, this is the case if and only if Sylvester's law of nullity holds with respect to the rank function, for maps between projective modules. Further, the ring must have weak global dimension at most two. This is the content of Sections 2 and 3.

The Universal Field of Fractions of a Semifir I. Numerators and Denominators

The Universal Field of Fractions of a Semifir I. Numerators and Denominators

Fields of Fractions for Group Algebras of Free Groups

Let KF be the group algebra over the commutative field K of the free group F. It is proved that the field generated by KF in any Mal’cev-Neumann embedding for KF is the universal field of fractions $U(KF)$ of KF. Some consequences are noted. An example is constructed of an embedding $KF \subset D$ into a field D with $D\;\not \simeq \;U(KF)$. It is also proved that the generalized free product of two free groups can be embedded in a field.

The free product of skew fields

In a recent paper [3] it was shown that the free product K * L of two fields (possibly skew) can be embedded in a field, and moreover, this latter can be chosen to be the ‘universal field of fractions’ of K*L (cf. [4, 5]). This opens up the prospect of doing for skew fields what the Neumanns and others have done for groups; indeed some sample applications were given in [3]. We pursue this topic here a little further: our main results state (i) every ccuntably generated field can be embedded in a 2-generator field, (ii) in a free product of rings over a field k, any element algebraic over k is conjugate to an element in one of the factors, (iii) any field can be embedded in a field with nth roots for each n. These results are all analogous to well known results in group theory (cf. [8]), and although the proofs are not just a translation of the group case, the latter is of scme help. Thus (ii), (iii) follow fairly easily, but they lead to other problems, still open, by replacing ‘free product of rings’ in (ii) by ‘field product of fields, and in (iii) replacing ‘nth roots’ by ‘roots of any equation’. On the other hand, (i) is less immediate, since ‘field products’ need to be used in the proof, and their manipulation requires some more technical lemmas.

The word problem for free fields

It has long been known that every free associative algebra can be embedded in a skew field [11]; in fact there are many different embeddings, all obtainable by specialization from the ‘universal field of fractions’ of the free algebra (cf. [5, Chapter 7]). This makes it reasonable to call the latter the free field; see §2 for precise definitions. The existence of this free field was first established by Amitsur [1], but his proof is rather indirect and does not provide anything like a normal form for the elements of the field. Actually one cannot expect to find such a normal form, since it does not even exist in the field of fractions of a commutative integral domain, but at least one can raise the word problem for free fields: Does there exist an algorithm for deciding whether a given expression for an element of the free field represents zero?Now some recent work has revealed a more direct way of constructing free fields ([4], [5], [6]), and it is the object of this note to show how this method can be used to solve the word problem for free fields over infinite ground fields. In this connexion it is of interest to note that A. Macintyre [9] has shown that the word problem for skew fields is recursively unsolvable. Of course, every finitely generated commutative field has a solvable word problem (see e.g. [12]).The construction of universal fields of fractions in terms of full matrices is briefly recalled in §2, and it is shown quite generally for a ringRwith a field of fractions inverting all full matrices, that if the set of full matrices overRis recursive, then the universal field has a solvable word problem. This holds more generally if the precise set of matrices overRinverted over the field is recursive, but it seems difficult to exploit this more general statement.