Finiteness Assumption Research Articles

Higher order corrections to current quark mass induced isospin violation

A conjecture in QCD states that if either the up or the down current quark mass is zero one should observe a 50% isospin violation in the vector form factor of the radiative pion decay. Its correctness hinges on an assumption of finiteness in the infrared mass limit, which is known to be true to lowest order. We show by an explicit calculation in perturbative QCD that it is not true to next order. The proof of non-renormalizability of the above-mentioned result then does not go through.

Unitaries in Simple Artinian Rings

Let R be a 2-torsion free simple artinian ring with involution*. The element u of R is said to be unitary if u is invertible with inverse u*. In this paper we shall be concerned with the subalgebras W of R over its centre Z such that uWu* ⊆ W, for all unitaries u of R. We prove that if R has rank superior to 1 over a division ring D containing more than 5 elements and if R is not 4-dimensional then any such subalgebra W must be one of the trivial subalgebras 0, Z or R, under one of the following extra finiteness assumptions: W contains inverses, W satisfies a polynomial identity, the ground division ring D is algebraic, the involution is a conjugate-transpose involution such that D equipped with the induced involution is generated by unitaries.

Differential operators and primary ideals

Let A be a (commutative) Noetherian ring containing a field k. Let PC A be a prime ideal and let K be the field of fractions of A/P. The main result of this paper states that every P-primary ideal of A is the set of zeros of some finite dimensional coalgebra of differential operators from A to K. Actually, we make an additional finiteness assumption. If A’ is the localization of A at P, we assume there is a field k’, K C k’ C A’, with K Or, A’ also Noetherian. This is perhaps more general than assuming K finitely generated over k’. In any event, there is no need to assume A finitely generated over a field. For example, A could be a quotient of a power series ring. The’ differential operators Diff,“,,(A, K) form a subspace of the K-linear (but not necessarily A-linear) maps Horn&A, K). The precise definition is given in Section 1. We have Diff,“,,(A, K) = Diff,“,,,(A’, K)1 Diff,“,,,,(A’, K). If K Ok, A’ is Noetherian, then .Q = Diff,“,,,,(A’, K) is a finite dimensional coalgebra over K. Roughly, this means given DE Diff,“,&A’, K), there is a formula for all a, b E A’, D(u . b) = X)‘(u) D”(b), for suitable D’, D” EDiff:.,,,(A’, K). It is relatively easy to prove that if & C 9 is a non-zero subcoalgebra, then Q(b) = {u E A / Da = 0 all D E &?} is a P-primary ideal, containing P”+l. The more interesting part of the theorem is the converse, that there are indeed enough subcoalgebras of operators to detect all primary ideals. If, in the notation above, K is a separable algebraic extension of k’ C A’, then, in fact, there is a bijective correspondence between subcoalgebras of Diffi”.,,, (A’, K) and P-primary ideals containing Pn+l. If the characteristic is p > 0 and the degree of inseparability of K over k’ is finite, say p”, then we still get a bijective correspondence between P-primary ideals containing Pn+l and subcoalgebras of Diffz,&A’, K), if th e s m o s are suitably interpreted. It is .y b 1 necessary to replace the obvious A’-module structure on A’ and K by the structure induced by the p” power map on A’, that is, a o x = upex.

Vanishing Homology over Nilpotent Groups

Let $\pi$ be a nilpotent group and let $M$ be a $\pi$-module. Under certain finiteness assumptions we prove that the twisted homology groups ${H_i}(\pi ,M)$ vanish for all positive $i$ whenever ${H_0}(\pi ,M) = 0$.

Vanishing homology over nilpotent groups

Let π \pi be a nilpotent group and let M M be a π \pi -module. Under certain finiteness assumptions we prove that the twisted homology groups H i ( π , M ) {H_i}(\pi ,M) vanish for all positive i i whenever H 0 ( π , M ) = 0 {H_0}(\pi ,M) = 0 .

On the Modeling of Demand Paging Algorithms by Finite Automata

Two finite automata are devised for modeling two classes of demand paging algorithms. The first one of one input and three outputs models the class of algorithms with a constant amount of allocated space. The second one of one input and six outputs models the class of algorithms with a variable amount of allocated space. Some evaluation techniques are developed following each model. The memory states of the first class algorithm with the Least Recently Used (LRU) replacement policy and the working set model of the second class are recursively defined by strings of the loaded pages. The adopted replacement policy and the state string updating procedure are imbedded in the recursive definition of memory states. Properties of some algorithms are developed to fit the finiteness assumption of a reference string.

On the Structure of Semiprime Rings

The structure of prime rings has recently been studied by A. W. Goldie, R. E. Johnson, L. Lesieur and R. Croisot. In their main results some sort of finiteness assumption is invariably made. It is shown in this paper that certain semiprime rings are subdirect sums of full rings of linear transformations of a right vector space over a division ring. No finiteness assumption is made about the ring. An apparently new radical property is defined and some of its properties are established; e.g., the radical of a matrix ring ${R_n}$ is the matrix ring of the radical of $R$.

On the structure of semiprime rings

The structure of prime rings has recently been studied by A. W. Goldie, R. E. Johnson, L. Lesieur and R. Croisot. In their main results some sort of finiteness assumption is invariably made. It is shown in this paper that certain semiprime rings are subdirect sums of full rings of linear transformations of a right vector space over a division ring. No finiteness assumption is made about the ring. An apparently new radical property is defined and some of its properties are established; e.g., the radical of a matrix ring R n {R_n} is the matrix ring of the radical of R R .

Finiteness assumptions and intellectual isolation of computer scientists

No abstract available.

A closure property of sets of vectors

Our aim here is to study this property. The exclusion of the vector 0 is not essential in what follows. It turns out that many of the other properties of root systems of Lie algebras depend only on property P and are thus shared by a much wider class of vector systems. In the statements to follow it is assumed that all vectors considered come from a real Euclidean space Vof finite dimension n. 1.1 Let S be finite and have property P, and let A be a subset with property P. For each x in S, assume that at most one (resp. exactly one, at least one) of x and -x is in A. Then there is an ordering of the space V such that A is contained in (resp. A is, A contains) the set of positive elements of S. Harish-Chandra [6, Lemma 4] proves the second part of this result for Lie algebra root systems, however, using nontrivial properties of Lie algebras. Borel and Hirzebruch [2, pp. 471-473] give a geometric proof of the second and third parts, but then revert to Lie algebra techniques to prove the first part, all for Lie algebra root systems. All of these authors make a somewhat stronger assumption on A than property P, namely, if x, y e A and x + y e S, then x + y e A. Our proof of 1.1 depends on a preliminary result which may have some independent interest. 1.2. Let B be finite and have property P, and assume that x e B implies -x q B. Then ther-e is an ordering of the space V such that all elements of B are positive. The real numbers of the form k 112 (k, I positive integers) show that the assumption of finiteness in 1.1 (or 1.2) can not be dropped. It can, however, be weakened thus. 1.1'. In 1.1 replace the assumption of finiteness by the assumption that 0 is not a point of accumulation of S. In 1.2 we can go further, in terms of a weakening of property P.

On the Galois theory of division rings

1. Throughout this paper, K will represent a division ring and L a galois division subring. We are interested in establishing a galois theory for the extension K/L when K/L is locally finite. In order to do this one must identify the galois subrings of K containing L. An example given by Jacobson [4] shows that not every such division subring is galois. However, we obtain that each subring subject to a natural finiteness assumption is galois when the dimension [K: H], <?o where H is the division subring left fixed by all inner automorphisms which leaves L fixed. For these subrings we then obtain the usual theorems. We rely strongly on topological characterizations of the galois groups. In particular, we characterize the extensions with locally compact and complete galois groups. These results provide extensions and simplications of the recent results of Nagahara, Tominaga, and Nobusawa on division rings cited in the bibliography.

A generalization of the ideal theory of commutative rings without finiteness assumptions

A generalization of the ideal theory of commutative rings without finiteness assumptions

On direct decompositions. II

The theory developed in our first paper is applied here to the case of algebras A such that every pair of congruences on A commute. The general refinement theorem which we obtain is already known when the ascending and descending chain conditions hold for the lattice of congruences. This result is due to O. Ore. We here weaken this finiteness assumption to a point comparable with that of group theory when both chain conditions are assumed for subgroups of the centre. Our path to the theorem is obtained by adapting to a different situation methods developed by Kurosch, Baer, and Jónsson and Tarski. From Kurosch (4) we take the concept of the centre of a pair of direct decompositions, which we extend and formulate as a pair of mappings π [α, β; γ, δ] over the set of congruences. In this connexion, Theorems 12·5 and 12·6 appear to be new, even for groups. From Baer (1) we take the concept of canonical refinements and generalize it to our case. The final approach (§ 14) to the refinement theorem is essentially following Jónsson-Tarski (3). Possibly our most interesting conclusion is that the elegant methods using endomorphisms, introduced by Fitting into group theory, generalize naturally to our case.

The Radical of an Alternative Ring

In this paper we shall show that N. Jacobson's definition of the radical of an (associative) ring [J1, B] applies to alternative rings [Z1, M], and we shall develop some of the elementary properties of this radical. The radical of an alternative ring was first discussed by M. Zorn [Z2] under certain finiteness assumptions. Dubisch and Perlis [D P] have more recently studied the radical of an alternative algebra (of finite order). We make no finiteness assumptions. We do, however, prove that the chain conditions employed by Zorn [Z2, (4.2.1)(4.2.3)] ensure that the radical defined by him coincides with that defined in this paper. Our discussion applies equally well to algebras of possibly infinite order [JI, ?6] so that our results essentially contain some of the results of Dubisch and Perlis. We have been unable to discover the relation between the radical and maximal right ideals, nor have we developed any parallel for Jacobson's structure theory of associative rings. The enlarged radical of Brown and McCoy [B-McC] will yield a type of structure theory for arbitrary non-associative rings, but because of the generality involved we prefer to leave a discussion of this interesting fact to a subsequent publication.

Homology of spaces with operators. I

The object of this paper is to study simultaneous invariants of a topological space X and of an abstract group W acting as a group of transformations on X. We assume that W also acts as a group of operators on the coefficient group G. In addition to the ordinary homology groups of X over G we define two ones that we call the equivariant and the residual homology groups. The three kinds of homology groups are connected by homomorphisms which form an exact sequence, that is, a sequence in which each homomorphism has as kernel the image of the preceding homomorphism. The same is done with cohomologies. We use singular homology theory [31(1) throughout; this permits us to drop all finiteness assumptions on W. In this respect our approach seems to differ from the one very successfully used by P. A. Smith [16]. In Chapter I we develop the theory of abstract complexes with operators. In Chapter II we define the special groups for a topological space X and show that if X is a simplicial complex and if W operates on X simplicially, then the special groups can be computed from the simplicial scheme of X. In the last part of Chapter II we define for each (discrete) group W an abstract complex Kw which has Was a group of operators. The equivarient groups of Kw are shown to be isomorphic with the homology and cohomology groups of the group W studied by Eilenberg and MacLane [5, 6, 7] and Hopf [10]. In Chapter III we assume that W operates on X without fixed points and that the ordinary homology groups of X vanish up to a certain dimension. It is then shown that the equivariant and residual groups of the space X are isomorphic with the appropriate groups of the group W. In Chapter IV these results are applied to prove some theorems about continuous mappings of spaces with operators. In Chapter V we take up the homology theory with local coefficient introduced recently by Steenrod [17]. Roughly speaking a homology (or cohomology) group with local coefficients arises whenever the fundamental group irl(X) operates on the coefficient group G. If Xdenotes the universal covering space of X, then W=7r1(X) operates both on Xand on G thus giving rise to the equivariant groups of X-. We show that the equivariant

Structure theory of simple rings without finiteness assumptions

It is the purpose of this paper to lay the foundations of a general theory of simple rings, both associative and non-associative. In part I we obtain the structure of simple associative rings that either contain minimal right ideals or contain maximal right ideals. In either case we obtain a realization of our ring 2f as a certain type of ring of linear transformations in a vector space over a division ring. If 2{ has a minimal right ideal this realization is essentially unique and we can prove a converse theorem that rings of linear transformations having certain properties are simple and contain minimal right ideals. Thus the theory here is essentially as complete as that of the classical case of rings that satisfy the descending chain condition for right ideals. Our structure theory hinges on a general theorem (Theorem 6) on the structure of irreducible rings of endomorphisms of commutative groups. This theorem may be regarded as an extension of Burnside's theorem on irreducible algebras of matrices. The classical theory of simple rings with descending chain condition is a simple consequence of our results and we believe that the present treatment is more transparent than the methods of proof previously given(1) . In studying an arbitrary non-associative ring 9t one is led to consider the associative ring 92 generated by the left and the right multiplications x->ax and x-xcxa acting in 2. If 21 is simple, 91 is an irreducible ring of endomorphisms. In part II we describe the structure of $1 in terms of the multiplication centralizer ( of 2[ defined to be the totality of endomorphisms 'y in 2 such that (xy)y=(xy)y=x(yy). We define the concepts of center, central algebra and extension of the underlying field of an algebra. In Part III we investigate a special type of simple associative algebra that may be regarded as a generalization of the concept of the complete algebra of linear transformations in a vector space over a field, or equivalently, of the concept of the complete matrix algebra. There are many points of contact between the discussion here (and in other parts of the paper) and recent work in the theory of rings of transformations in Banach spaces and in vector spaces over the fields of reail and complex numbers(2).