Showing papers on "Ring (mathematics) published in 1970"

The value-semigroup of a one-dimensional Gorenstein ring

Abstract: In a conversation about [4], 0. Zariski indicated to the author that there should be a relation between Gorenstein rings and symmetric value-semigroups, possibly allowing a new proof for a result of Herzog on complete intersections. In the following note it is shown that this is the case. We use the following facts about Gorenstein rings: If R is a onedimensional noetherian local ring with maximal ideal m and full ring of quotients Q(R), then the following conditions are equivalent: (a) R is a Gorenstein ring (by definition: m contains a nonzero divisor, which generates an irreducible ideal). (b) Each principal ideal, generated by a nonzero divisor, is irreducible. (c) The length of the R-module m-'/R is 1. (d) For each ideal a of R, which contains a nonzero divisor, (a-')-'=a. Here the inverse of an ideal is taken in Q(R). For easy proofs of these equivalences see Berger [3]. Let R be the integral closure of R in Q(R) and f the conductor from R to W. If W is a finitely generated R-module, then f contains a nonzero divisor. If R is Gorenstein, then the length of the R-module R/f is 2d, where d is the length of R/f. Roquette [5] gives the following simple proof: If

Near-rings in which each element is a power of itself

Abstract: Let R denote a near-ring such that for each x ∈ R, there exists an integer n(x) > 1 for which xn(x) = x. We show that the additive group of R is commutative if 0.x; = 0 for all x ∈ R and every non-trivial homomorphic image R¯ of R contains a non-zero idempotent e commuting multiplicatively with all elements of R¯. As the major consequence, we obtain the result that if R is distributively-generated, then R is a ring – a generalization of a recent theorem of Ligh on boolean near-rings.

Homological properties of the ring of differential polynomials

Abstract: The ring of differential polynomials over a universal differential field (Kolchin [7]), and the ring of twisted polynomials F2[), p] , where F2 is an algebraic closure of Z /2Z and p is the automorphism of JF2 defined by: &->z , "localized" at the multiplicative subset {t\k an i n t e g e r ^ 0 } , provide examples of a principal right and left ideal domain R, not a field, that is a right F-ring (i.e., each simple right i£-module is injective). Such a ring was conjectured to exist by Carl Faith. Both examples are shown to have a unique simple right -R-module. If R is either example, then by definition of a right F-ring, every right i?-module has a maximal submodule. Bass proved that if a ring A satisfies the d.c.c. on principal left ideals, then A has a bounded number of orthogonal idempotents and every right Amodule has a maximal submodule. The above examples show that the converse is false, thus answering a question raised by Bass [l, p. 470]. The author is deeply indebted to his adviser, Professor Carl Faith, for suggesting the problems treated in this paper and for his generous advice and encouragement.

Decomposability of finitely presented modules

Abstract: It is proved that a commutative ring with 1 has the property that every finitely presented module is a summand of a direct sum of cyclic modules if and only if it is locally a gen- eralized valuation ring. A Noetherian ring has this property if and only if it is a direct product of a finite number of Dedekind do- mains and an Artinian principal ideal ring. Any commutative local ring which is not a generalized valuation ring has finitely presented indecomposable modules requiring arbitrarily large

The chern-dold character in cobordisms. i

Abstract: The fundamental results of the paper are: derivation of formulas for the Chern-Dold character chU and chU in the theory of universal bordisms U* and cobordisms U*, respectively; derivation of the formula of a series over the ring Ω*U which gives addition in the formal group of "geometric" cobordisms, and derivation of the formula for the series kΨUk(u), where uU2() and the ΨkU are Adams operators in U*-theory. Bibliography: 13 items.

The semigroup of endomorphisms of a Boolean ring

Abstract: The family (R) of all endomorphisms of a ring R is a semigroup under composition. It follows easily that if R and T are isomorphic rings, then (R) and (T) are isomorphic semigroups. We devote ourselves here to the converse question: ‘If (R) and (T) are isomorphic, must R and T be isomorphic?’ As one might expect, the answer is, in general, negative. For example, the ring of integers has precisely two endomorphisms – the zero endomorphism and the identity automorphism. Since the same is true of the ring of rational numbers, the two endomorphism semigroups are isomorphic while the rings themselves are certainly not.

Homological dimension and cardinality

Abstract: Let {e(i) I i E f} be an infinite set of commuting idempotents in a ring R with 1 such that n m 1l e(ia) lJ (1-e(ia)) # 0 a=O 13=n+l for {ia I 0 M,,, then any right and left self-injective ring which is not semiperfect, any ring containing an infinite direct product of subrings, any ring containing the endomorphism ring of a countable direct sum of modules, and many quotient rings of such rings must all have infinite global dimension. This paper continues the investigation of the relationship between homological dimension and cardinality questions started in [4] and [5], combining the techniques of ?7 of [5] with a modification of a projective resolution in Pierce [7]. Employing a result of Hausdorff and Tarski, we show that the X corresponding to 2To plays an important role in the global dimension of rings where one can find analogues of characteristic functions of subsets of a set of orthogonal idempotents. 1. Homological dimension of an ideal generated by commuting idempotents. We first calculate the homological dimension of a right ideal of a ring R (with 1) generated by a "nice" set of idempotents, and then show that several types of rings possess such idempotents. A family t = {e(i) I i E f} of idempotents of R is called nice if (i) e(i)e(j) = e(j)e(i) Vi, j Ef . (ii) Ila= 1 e(ia) 171 n +1 (1-e(i))#O if {ia 1 R c R_n io < i < *-in Received by the editors December 1, 1969. AMS Subject Classifications. Primary 1690; Secondary 0430, 1640.

An embedding theorem for HCF-rings

Abstract: Let D be an integral domain with identity. In (3), Cohn has shown that if D is a Schreier ring, then D can be inertly embedded in a pre-Bezout domain B(D), which Cohn calls the pre-Bezout hull of D. Further, D is an HCF-ring if and only if B(D) is a Bezout domain, and D is a UFD if and only if B(D) is a PID. Using a modification of Cohn's techniques, Samuel in (8) has shown that a Krull domain can be inertly embedded in a Dedekind domain. In this paper, we show that these embedding theorems are also obtained by considering the embedding of a v-ring in its Kronecker function ring with respect to the v-operation. To wit, if D is a v-ring and if Dv is the Kronecker function ring of D with respect to the v-operation, then Dv is a Bezout domain, and we show that D is inertly embedded in Dv if an only if D is an HCF-ring. Moreover, D is a UFD if and only if Dv is a PID, and if D is a Krull domain, D is inertly embedded (in Samuel's sense) in Dv, a Dedekind domain.

On a conjecture of samuel

Abstract: We construct examples disproving Samuel's conjecture stating that the ring A[[T]] is factorial for a complete factorial local ring A. We also prove a theorem asserting (under some restrictions) that the ring A[[T]] is factorial for a geometrically factorial ring A. The bibliography contains 16 items.

On the Nilpotency of Nil Subrings

Abstract: A famous theorem of Levitzki states that in a left Noetherian ring each nil left ideal is nilpotent. Lanski [5] has extended Levitzki's theorem by proving that in a left Goldie ring each nil subring is nilpotent. Another important theorem in this area which is due to Herstein and Small [3] states that if a ring satisfies the ascending chain condition on both left and right annihilators, then each nil subring is nilpotent. We give a short proof of a theorem (Theorem 1.6) which yields both Lanski's theorem and Herstein- Small's theorem. We make use of the ascending chain condition on principal left annihilators in order to obtain, at an intermediate step, a theorem (Theorem 1.1) which produces sufficient conditions for a nil subring to be left T-nilpotent.

Rings with a discrete group of divisor classes

Abstract: We shall say that a ring A has a DGC (discrete group of classes) if the group of divisor classes is preserved in going to the ring of formal power series, i.e. C(A) → C(A[[T]]) is an isomorphism. We prove the localness and faithfully flat descent of the DGC property. We establish a connection between the DGC property of a ring and its depth. We also give a characterization of two-dimensional rings with DGC and characteristic zero rings with DGC. Finally, it is shown that the discreteness of the group of divisor classes is preserved under regular extensions of rings such as A[T1,⋯, Tn], A[[T1 ,⋯, Tn ]], completions, etc. Bibliography: 13 items.

The virial theorem applied to magnetospheric dynamics

Abstract: The application of the virial theorem to the magnetosphere has been extended to the study of a closed magnetosphere. The result is used to express the field at the earth due to external current systems in terms of various self energies of the external systems. The special case of an isolated ring current reduces to the usual result. A new expression is obtained for the boundary current contribution and for the combined ring and boundary currents. Energy considerations are given that lead to an estimate of the size of the ring current effect on geomagnetic sudden impulses and sudden commencements.

Injective Endomorphisms of Finitely Generated Modules

Abstract: Let R be a commutative ring. Then any injective endomorphism of a finitely generated R-module is always an isomorphism if and only if R is 0-dimensional, that is, if every prime ideal is maximal. This note aims at considering cases where an injective endomorphism of a finitely generated module is, actually, an isomorphism. It is a simple exercise that artinian modules are endowed with this property [1, p. 23] and here we will show that the commutative rings for which the fact above is always true resemble artinian rings. A similar question on when surjective endomorphisms of finitely generated modules are isomorphisms was proved independently by Strooker [3] and the author [4] or [5] for any commutative ring, regardless of finite presentation [2, p. 35 ] or chain conditions [1, p. 23]. For a commutative ring R, the result of this note says THEOREM. Any injective endomorphism of a finitely generated Rmodule is an isomorphism if and only if every prime ideal of R is maximal. That the above condition is necessary, it is easy to see: If PCQ are two distinct primes in R, then any element xEQ-P induces, via multiplication, an injection of R/P which is not surjective. The converse takes longer to prove but it is just as easy. Consider thus a ring R with the aforementioned property, that is, of having Krull dimension 0, and let f be an injection of the finitely generated R-module M. This module can be made into an R[x]module by defining x m=f(m) for m M. We claim that as an R [x ]-module M has an annihilator I, big enough, so that S = R [x ]/I is 0-dimensional. To see this, let ml, * * *, m1, be a generating set for M as an R-module. We have xmi= Ermi,m with rijCR, that is, a system of equations Received by the editors February 6, 1970. AMS Subject Classifications. Primary 1320; Secondary 1340.

Preparation of X-ray negatives of tree-ring specimens for dendrochronological analysis.

Abstract: Techniques for producing X -ray negatives of dendrochronological specimens have been developed at the Geological Survey of Canada and the Nondestructive Testing Laboratory, Mines Branch, The radiographs are produced to provide tree -ring density data to supplement ring -width measurements for dating and climatic studies. New specimen preparation techniques and X -ray methods are discussed. The quality and quantity of tree -ring information is enhanced by the use of X -ray analysis. INTRODUCTION The usefulness of tree -ring density analysis for evaluating the commercial quality of wood and for dendroclimatic interpretation has been demonstrated in recent years by Polge (1965a, 1965b, 1966), Green and Worrall (1964), Green (1965), and Harris (1969). One of the best techniques for obtaining graphic and quantitative tree -ring density data is to scan X -ray negatives of dendrochronological specimens on a densitometer (Polge 1966; Jones and Parker 1970). The purpose of this paper is to describe the technique used at the Geological Survey of Canada to prepare X -ray negatives of tree -ring samples for dendrochronological analysis. Particular attention is given to the production of radiographs that can be used for the two traditional foci of dendrochronology: (1) the dating of tree -ring specimens, and (2) the relating of tree -ring data to climatic factors. Accurate crossdating of the annual rings in tree -ring series and the assigning of correct calendar year dates to each ring in tree -ring chronologies is essential for both dendrochronological dating and dendroclimatic analysis. Tree -ring density measurements obtained from X -ray negatives can be used to supplement ring -width measurements for crossdating purposes (Jones and Parker 1970). Tree -ring chronology characteristics such as false annual rings, locally absent rings, rings with very faint latewood, and microscopic rings are important considerations in crossdating` ring widths. These same characteristics need to be considered in tree -ring density studies and X -ray examples of these features are presented. Techniques in field collection and specimen preparation used at the Geological Survey of Canada are designed to accommodate both tree -ring density and tree -ring width analysis. Preparation requirements are more stringent for density studies and new specimen preparation methods have been devised. Specimen preparation and X -ray exposure procedures are designed to accommodate large quantities of material. Two techniques for producing the X -ray negatives have been developed. In one method the X -ray film, the tree -ring specimen, and the X -ray source are held stationary during exposure. This technique is similar to that used by Polge (1966) in his tree -ring density studies. In the other method the X -ray source is held stationary and emits radiation through a narrow slit onto a moving carriage supporting the tree -ring specimen and film. 12 TREE -RING BULLETIN

Computable fields and arithmetically definable ordered fields

Abstract: Introduction. A computable field is one whose elements may be placed in one-one correspondence with the natural numbers in such a way that the number theoretic functions corresponding to the field operations are recursive. In the same vein a field is called arithmetically definable (AD for short) if its elements may be placed in one-one correspondence with the natural numbers in such a way that the number theoretic functions corresponding to the field operations are arithmetical. These notions clearly extend in an obvious way to ordered fields and indeed to algebraic structures in general. The term computable structure (group, ring, etc.) was probably introduced for the first time by M. 0. Rabin [4], however, a similar notion was discussed a few years earlier by Frohlich and Shepherdson [1]. Each of these references contains a number of interesting theorems on computable structures. Some results concerning AD structures appear in [2 ]. The main purpose of the present paper is to show that the fields of real algebraic numbers, constructible numbers, and solvable numbers, which were shown to be AD in [2], are in fact computable. This answers a question raised in footnote (2) of [2 ].

A noncommutative Hilbert basis theorem and subrings of matrices

Abstract: A finitely generated central extension Aul,. ,Uk] of a commutative noetherian ring A, satisfies the ascending chain condition for ideals P for which Atul,. . ., Ukl/P can be embedded in matrix rings Mn(K) over arbitrary commutative rings K and n bounded. The method of proof leads to an example of a ring R which satisfies the same identities of Mn(K) but nevertheless cannot be embedded in any matrix ring over a commutative ring of arbitrary finite order.

On semigroups admitting ring structure

Abstract: The set of elements of an (associative) ring under multiplication form a semigroup, but not every semigroup is isomorphic to the multiplicative structure of a ring. The class of all multiplicative semigroups of all rings can not be described axiomatically. Nevertheless for unique-addition rings, for finite rings and other special cases interesting characterizations can be given.

Balanced rings and a problem of Thrall

Abstract: Balanced ring is defined and related to Thrall's QF-1 rings. Several theorems are obtained which show that balanced rings enjoy strong homological and chain conditions. The structure of commutative balanced rings is determined. Also, the structure of commutative artinian QF-1 rings is gotten. This is a generalization of a theorem of Floyd. Introduction. If M is a right R-module, then M is a natural left module over its endomorphism ring S. We call T= Ends M the BiEndomorphism ring of M, and the elements of Tare called BiEndomorphisms (notation BiEnd M). The mapping: -q: R -> BiEnd MR, ?: a-> ad, where (x)ad = xa, Vx E M is a ring homomorphism and ker 71 is the annihilator of M (notation ker Xj = annR M). The elements of BiEnd MR of the form ad are called right multiplications of M. Every element of BiEnd MR is a right multiplication iff the natural map D: R -? BiEnd M is surjective, that is, a ring epimorphism. In this case, following Faith [2], we say M is balanced. If M is balanced, we have a complete description of BiEnd MR, namely BiEnd MRR/annR M. Naturally, this is not always the case, as is well known. In this paper we study rings for which every right R-module is balanced, and call R right balanced in this case. It appears that balanced tings have not been studied in this generality. Thrall [13] proposed the classification of finite dimensional algebras, called QF-1 algebras, having the property that every finitely generated faithful right R-module is balanced. The problem remains unsolved at the present, but there are results in special cases (Floyd [5], Fuller [7] and Morita [9]). The point of departure of this paper, and the idea which led to our main results is the observation that first, the QF-1 hypothesis, when assumed for general rings and their quotients, actually implies chain conditions, and second that the determination of BiEnd MR is what we want, for a general R-module M, not merely Received by the editors September 8, 1969. AMS Subject Classifications. Primary 1340, 1350.

The annihilator of radical powers in the modular group ring of a $p$-group

Abstract: We show that if N is the radical of the group ring and L is the exponent of N, then the annihilator of NW is NL-W+1. As corollaries we show that the group ring has exactly one ideal of dimension one and if the group is cyclic, then the group ring has exactly one ideal of each dimension. This paper deals with the group ring of a group of prime power order over the field of integers modulo p, where p is the prime dividing the order of the group. This field is written as K and the group ring as KG. It is well known that KG is not semisimple; if N is the radical of KG and NL 5 0while NL+1 =0, then L is said to be the exponent of N. We prove the following result: THEOREM. Let G be a p-group and KG be the group ring of G over K = GF(p), the field with p elements. If L is the exponent of the radical, N, of KG, then the annihilator of Nw is NL-W+1. For S a nonempty subset of G, let S+ = iE s gi; in particular, for H a normal subgroup of G, let (H+) be the ideal in KG generated by H+. For g and h in G, the following identities are used: (g 1)P-1 = 1 + g + g2 + . + gp-; (g 1)P = gP 1; (gh 1) = (g 1)(h 1) + (g 1) + (h 1); and (h-1)(g-1) = (g-1)(h1) + (gh-1)(c-1) + (c-1) where c = (h, g) = h-g-lhg. The following definitions and theorems are due to Jennings [1]. Let Ki be the set of all elements gi in G such that gi 1 mod Ni. THEOREM 1 [1, THEOREM 2.2]. The sets Ki, i=1, 2, * * , form a decreasing series of characteristic subgroups of G. This series of subgroups will be referred to as the K-series of G. THEOREM 2 [1, THEOREM 2.3]. The K-series of any group G has the following properties: Presented to the Society, April 18, 1970; received by the editors January 16, 1970. AMS Subject Classifications. Primary 2080; Secondary 1630.

Konservative Abbildungen und Fortsetzung regulÄrer Maße

Abstract: Finite inner regular measures (not necessarily assumed to be σ-additive) on a ring are shown to be the maximal elements with respect to a certain order relation on the set of all finite measures on the ring. By means of this result we prove a characterization of those regular measures which are the images of regular measures under a given measurable transformation. Generalizing certain results of H. Bauer, applications are given to continuous mappings of Hausdorff spaces into Hausdorff spaces.

Rings of polynomials

Abstract: For aii algebra R over a field k, with residue field K to be a ring of polyniomials in one variable over k it is necessary that trdeg K/k = 1. We prove that under the hypothesis tr* deg K/k -1, R is a ring of Krtull-dimension at most one. This is used to derive sufficient conditions for R to be a ring of polynomials in one variable over k. 1. Let k be a subfield of the commutative ring R. Let K be the quotient field of R. The problem we are concerned witlh is: When is R a ring of polynomials? IIn a previous paper [1] we obtained the following result: If R is a subring of k [Xi . . . Xn] such that with every element of R all of its factors in k [XIc xn] already lie in R, and if tr deg K/k =n, then R is a ring of polynomials. One of the results that we get in this paper is that R is a ring of polynomials also in case tr deg K/k = 1. We start by studying rings R for wlhich tr deg K/k_? 1. We prove that if R is a unique factorization domain, and R is a subring of the ring of polynomials k [xI . . . Xn], then R is a ring of polynomials. For subrings of the rings of polynomials over k we prove that (i) if R is a principal ideal domain then R is a ring of polynomials, and (ii) if R has a principal ideal M so that R/M is canonically isomorphic to k, then R is a ring of polynomials. Some possible generalizations and modifications are also pointed Dut. 2. The main object of this section is the study of the rings R for which tr deg K/k? 1. THEOREM I. If kCR, and if tr deg K/k ?1, then Krull-dim R 1, and let us lerive a contradiction. Since there exist prime ideals P, Q in R so that Received by the editors November 12, 1969. A IS Subject Classifications. Primary 1393; Secondary 1230, 1420. Key WVords antd Phrases. Rings of polynomials, rings of power series, unique facorization domain, principal ideal domain, Euclidean domain, transcendence degree, Irull dimension.

HOMOLOGICAL DETERMINACY OF p-ADIC REPRESENTATIONS OF RINGS WITH POWER BASIS

Abstract: For rings with power basis it is proved that if two modules which are free and finitely generated over the ring of p-adic integers have cohomology of identical behavior, then they are "almost isomorphic."

Commutative qf-1 artinian rings are qf

Abstract: In a recent paper, D. R. Floyd proved several results on algebras, each of whose faithful representations is its own bicommutant (=R. M. Thrall's QF-1 algebras, a generaliza- tion of QF-algebras) among which was the theorem in the title for algebras. We obtain our extension of Floyd's result by use of inter- lacing modules, replacing his arguments involving the representa- tions themselves. In (10), Thrall observed that the class of finite-dimensional algebras over which every faithful representation has the double centralizer property (i.e., is its own bicommutant) properly contains the class of quasi-Frobenius ( = QF) algebras. He called the members of the former class QF-1 algebras and posed the intriguing problem of characterizing these algebras in terms of ideal structure. Solutions for this problem have been given for generalized uniserial algebras (5) and for commutative algebras (4). Every faithful module M over a QF ring R has the double centralizer property in the sense that the natural homomorphism X: R--Homc(M, M) (where C= HomR(M, M)) is onto (see (2, ?59)). Thus Thrall's definition and his problem extend naturally to QF-1 artinian rings. In view of recent results on the dominant dimension of an artinian ring (see (1, Theorem 2) or (8, Lemma 9)), the characterization of QF-1 generalized uniserial algebras (and its proof) given in (5) re- mains valid for generalized uniserial rings. In this note we prove the theorem of the title, thus extending a theorem that Floyd proved for finite-dimensional algebras by means of matrix representations (4). It is not difficult to show that a direct sum of rings is QF or QF-1 if and only if so is each of the direct summands. Thus for our purposes we may assume that R is a commutative local artinian ring. Accord- ing to Nakayama's original definition (9, p. 8), such a ring is QF if and only if its R-socle (i.e., its largest semisimple R-submodule) S(R) = S(RR) is simple. We shall prove the theorem by constructing, in the event that S(R) is not simple, a faithful module whose double cen- tralizer has an R-socle larger than that of R. The methods used in this

Primitive near-rings

Abstract: The theory of near-rings has arisen in a variety of ways There is a natural desire to generalise the theory of rings and skew fields by relaxing some of their defining axioms It has also been the hope of some mathematicians that certain problems in group theory, particularly involving permutation groups and group representations, may perhaps be clarified by developing a coherent algebraic theory of near-rings Moreover, there is an increasing recognition by mathematicians in many branches of the subject, both pure and applied, of the ubiquity of near-ring like objects The first steps in the subject were taken by Dickson and Zassenhans with their studies of 'near-fields', and by Wielandt with his classification of an important class of abstract near-rings Papers by Frohlich, Blackett, Betsch and Laxton developed the theory considerably Lately authors such as Beidleman, Ramakotaiah, Tharmanatram, Maxson, Malone and Clay have all added to our knowledge The history of the subject has been strongly influenced by our knowledge of ring theory, and although this has often been beneficial it must not be overlooked that a number of important problems in near-ring theory have no real parallel in the theory of rings It is probably best to try to preserve a balance, and not to endeavour exclusively, either to generalise theorems from ring theory irrespective of their usefulness, or to ignore the theory of rings and attempt to formulate a completely independent theory In many cases our results are generalisations of theorems from ring-theory but at certain important junctures we will explicitly use the fact that we are dealing with a near-ring which is not a ring This is a very interesting development in the subject We proceed, in the first chapter, with a review of the terms and notation that will be used in this thesis Where definitions and concepts are of a specialized or technical nature and only used in one section, it seems more sensible to postpone introducing them until a more natural point in the proceedings Chapter 2 gives a summary of the results on the various radicals corresponding to the Jacobson radical for associative rings Most of these results are well known and readily available in the literature We also consider near-rings with one, or more, of these radicals zero We defined, in Chapter 1, three different types of primitive near-ring, which are all genuine generalisations of the ring theoretic concept Of these three, the two most important are 2-primitive and 0-primitive near-rings In Chapter 3, we examine 2-primitive near-rings with certain natural conditions imposed on them A theorem is obtained which could be considered to be the equivalent result for near-rings of the theorem classifying simple, artinian rings, due originally to Wedderburn and redeveloped by Jacobson Chapters 4 and 5 deal with 0-primitive near-rings satisfying certain conditions Chapter 5 is a generalisation of Chapter 4, but we felt that the mathematical techniques involved would be clearer if the special case in Chapter 4 was expounded first In these two chapters we classify a sizeable class of 0-primitive near-rings with identity and descending chain condition on right ideals Several types of prime near-rings have been developed in the literature In Chapter 6 we examine these and related concepts In the theory of rings, Goldies' classification of prime and semi-prime ring with ascending chain conditions, has been of immense importance Whether such a result could be obtained in the theory of near-rings is a matter for conjecture, at the moment We have made a start on the problem with the construction of a class of near-rings which behave in a very similar way to Prime rings with the Goldie chain conditions This is the content of Chapter 7 The inspiration for its came mainly from the proof of Goldies' first theorem, due to C Procesi, which is featured in Jacobson's book (Jacobson [1]) Chapter 8, is an attempt to initiate the development of a theory of vector groups and near-algebras which would play an important role-in the future theory of near-rings, in a way, perhaps, similar to the Ale vector spaces and algebras play in ring theory This may lead, in time, to results on 2-primitive near-rings with identity and a minimal right ideal, for example, or a Galois theory for certain 2-primitive nearrings For the former problem, the experience of the semi-group theorists (Hoehake [1] etc ) may prove useful Finally a note on the numbering of results and definitions etc If a reference is made, containing only two numbers, e g 112 then this means, "item 12 of section 1 of the present chapter" If a reference reads: 3112, then this means "item 12 of section 1 of Chapter 3

Embeddings in division rings

Abstract: A method for embedding a certain class of integral domains in division rings is devised. Integral domains A are constructed with a generalized valuation into a (noncommutative) totally ordered semigroup that need not be discrete. Then the multiplicative semigroup A\{0} is expressed as an inverse limit of semigroups each of which is embeddable in a group. Thus A\{0} can be embedded in a group G. The main problem is to introduce addition on G in order that G becomes a division ring by the use of eventually commuting maps of inverse limits. Introduction. Suppose that the multiplicative semigroup A* of a noncommutative integral domain A can be embedded in a group G. Some recent surprising discoveries show that there exist rings A for which an embedding A*CG is possible, but such that for any embedding of A* into any group G whatever, addition cannot be extended to all of G u {0} in order to obtain an embedding of A into a division ring ([1] and [6]). Under certain appropriate additional hypotheses on an integral domain with a valuation into the integers, P. M. Cohn embeds A c G u {0}, introduces a group topology on G, then defines addition on the subset A*A*-l which happens to be dense in G, and then finally extends addition to all of G [2]. Recent interest in the subject ([1], [6], and [3]), as well as the fact that treatises on ring theory find it necessary to quote this result [5, p. 257] but do not prove it because existing proofs are too complicated, are two reasons why a purely algebraic proof which avoids topology altogether is needed. The first objective of this note is to introduce a much wider class of rings than simply integral domains with an integer valued valuation. The second aim is to prove that any ring in this wider class can be embedded in a division ring. One of the main factors contributing to the length and complexity of the present proof is that in place of the integers, the range of the valuation is in a not necessarily commutative semigroup r. One of the appealing features of the present development is that if F is specialized to be commutative, our proof simplifies considerably. Furthermore, it involves no topological considerations of any kind whatsoever. 1. Rings with a noncommutative valuation. In an ordered ring, the valuation Received by the editors February 24, 1969 and, in revised form, November 25, 1969. AMS Subject Classifications. Primary 1646, 1615; Secondary 2092.