Showing papers on "Ring (mathematics) published in 1968"

Functors of Artin rings

Abstract: 0. Introduction. In the investigation of functors on the category of preschemes, one is led, by Grothendieck [3], to consider the following situation. Let A be a complete noetherian local ring, ,u its maximal ideal, and k = A/l the residue field. (In most applications A is k itself, or a ring of Witt vectors.) Let C be the category of Artin local A-algebras with residue field k. A covariant functor F from C to Sets is called pro-representable if it has the form

Noninjective cyclic modules

Abstract: In [3], it is shown that a ring R such that every cyclic right Rmodule is injective must be semisimple Artin. In this note, that proof is greatly simplified, and it is shown that a hereditary ring cannot contain an infinite direct product of subrings. R will denote a ring with 1, all modules will be unital right Rmodules, and all homomorphisms R-homomorphisms. For a module M, E(M) will denote its injective hull (see [2]).

Reflexive modules over Gorenstein rings

Abstract: Introduction. The aim of this paper is to show the relevance of a class of commutative noetherian rings to the study of reflexive nmodules. They include integrally closed domains, group algebras over these, and Gorenstein rings. We will be basically concerned with a ring R having the following property: Let M be a finitely generated R-module; then M is reflexive if and only if every R-sequence of at most two elements is also an M-sequence. In the rest of this note, using the above characterization, we examine the closeness between free modules and Macaulay modules of maximum dimension in a local Gorenstein ring. Finally, it is proved that over a one-dimensional Gorenstein ring only projective modules have projective endomorphism rings.

Rings with restricted minimum condition

Abstract: Definition. A ring R is said to satisfy the restricted minimum condition (or to be a RM ring, for short), if for each ideal A $ (0) in R, the ring R/A is right artinian. In this paper we consider a RM ring, and are furthermore interested in the case where R itself is not right artinian. The concept of a commutative RM ring was introduced by I. S. Cohen [1], who also proved the following results for a commutative ring R: (a) R is RM iff R is noetherian and every proper prime ideal is maximal. (b) R is RM but not artinian iff R is a noetherian integral domain not a field, in which every proper prime ideal is maximal.

A presentation of SL 2 for Euclidean imaginary quadratic number fields

Abstract: Let G be any group and G′ its derived, then G/G′ —the group G made abelian—will be denoted by G a . Over any ring R , denote by E 2 ( R ) the group generated by the matrices as x ranges over R ; the structure of E 2 ( R) a has been described in a recent theorem [2; Th. 9.3] for certain rings R , the “quasi-free rings for GE 2 ” ( cf. §2 below). Now over a commutative Euclidean domain, E 2 ( R ) is just the special linear group SL 2 ( R ); this suggests applying the theorem to the ring of integers in a Euclidean number field. However, the only number fields whose rings were shown to be quasi-free for GE 2 in [2] were the non -Euclidean imaginary quadratic fields. In fact that leaves the application of Th. 9.3 of [2] to the ring of Gaussian integers unjustified (I am indebted to J.-P. Serre for drawing this oversight to my attention). In order to justify this application one would have to show either (a) that the Gaussian integers are quasi-free for GE 2 ., or (b) that Th. 9.3 of [2] holds under weaker hypotheses which are satisfied by the Gaussian integers. Our object in this note is to establish (b)–indeed our only course, since the Gaussian integers turn out to be not quasi-free.

The prime radical in a Jordan ring

Abstract: There are several definitions of radicals for general nonassociative rings given in literature, e.g. [1], [2], and [5]. The u-prime radical of Brown-McCoy which is given in [2], is similar to the prime radical in an associative ring. However, it depends on the particular chosen element u. The purpose of this paper is to give a definition for the Brown-McCoy type prime radical for Jordan rings so that the radical will be independent from the element chosen. Let J be a Jordan ring, x be an element in J; the operator U, is a mapping on J such that yU3=2x.(x.y)-x2.y for all y in J, or, equivalently, U.=2R!-R . If A, B are subsets of J, A UB is the set of all finite sums of elements of the forin aUb, where a is in A and b is in B.

Convergent higher derivations on local rings

Abstract: L. Introduction. In this paper we define a quasi-local ring R, or (R, M), to be a commutative ring with unity having a unique maximal ideal M such that nn=1 M ={O}. Thus a Noetherian quasi-local ring is a local ring. A higher derivation D = {Dj} 1 on a quasi-local ring R is said to be convergent if, for all a in R, :E?= 0 D,(a) is a convergent series in the M-adic topology. Do always denotes the identity mapping. If R is complete the mapping ocD: a -i= . Di(a) is an endomorphism of R which induces the identity mapping on the residue field of R (Lemma 1). With suitable restrictions on D, a,D is an automorphism and hence an inertial automorphism. A seemingly "natural" additional condition sufficient to insure that a,D is an automorphism is the condition

Inductive ring topologies

Abstract: Since topological algebra is the study of algebraic structures with topologies for which the operations are continuous, a natural question for the topological algebraist to ask is whether a given structure admits any such topologies whatever, other than the discrete and indiscrete ones. The question has been answered for some classes of structures. For example, Kertesz and Szele [7] prove that every infinite abelian group admits a nondiscrete, Hausdorff group topology. On the other hand, Hanson [5] gives an example of an infinite groupoid which admits only the two trivial topologies mentioned above. Our purpose here is to answer this question for infinite fields, proving that every infinite field admits a nondiscrete, Hausdorff field topology. This will be done by affirmatively answering the question for two classes of commutative rings: the first being all integral domains with a certain cardinality condition (§3), and the second, all rings which are the union of a chain of subrings with certain properties (§4). These two classes will be shown to include all infinite fields (§5). Our method of proof will make use of an inductive procedure first used by Hinrichs [6] to prove the existence of certain unusual topologies on the integers. The procedure is described in §1, where we define what we mean by an "inductive ring topology". In §§7 and 8, we turn our attention to some further applications of inductive topologies, showing first how they can be used to construct interesting examples of topologies on the integers and rational numbers. We use them to get proofs that there are uncountably many, and non-first countable ring topologies on all the rings considered in §3 and §4. We also show how characterizations can be obtained for several classes of topologies on fields using modifications of the inductive method. A supplement to our discussion of field topologies comes in §6, where we characterize those fields which admit nondiscrete, Hausdorff, locally bounded topologies. The methods used here, however, are those of valuation theory. When we say that a topology S~ is a ring topology on a ring A, we mean that the mappings (a, b)^-a—b and (a, b)-*ab from Ax A into A are continuous. &~ is a field topology on a field K if it is a ring topology, and in addition, the mapping a -*■ a'1 is continuous on K~{0}.

Totally integrally closed rings

Abstract: All rings and algebras in the following will be assumed commutative and associative. All rings will have identities, and ring homomorphisms will map the identity onto the identity. The identity of a subring will always be that of the ring. We recall that if A is a subring of a ring B, then b CB is said to be integral over A if there is a unitary polynomial f CA [X] such that f(b) = 0. B is said to be an integral extension of A if each b CB is integral over A. If B is an integral extension of A, then for every prime ideal (P of A there is a prime ideal Q of B with QnA = (P. If B is any extension of the ring A, the set A' of b CB which are integral over A is a subring of B and so an integral extension of A. If A =A', A is said to be integrally closed in B. If A CB CC where C is a ring and A and B are subrings, then if C is an integral extension of B and B an integral extension of A then C is an integral extension of A. From this it follows that the integral closure A' of A in an extension B is integrally closed in B. In the category of fields, integral extensions are just the algebraic extensions. The algebraically closed fields Q are precisely those which are injective with respect to algebraic extensions, i.e. they have the property that if u: KQ is a homomorphism where K is a field and E is an algebraic extension of K, then there is a homomorphism E-)Q agreeing with a on K. Each field K has an algebraic extension Q which is algebraically closed. Furthermore, if Q' is another algebraically closed algebraic extension of K, then any k-homomorphism QQ-2' (which always exists) is an isomorphism. The question naturally rises whether we get the analogous result in the category of rings when we consider integral extensions. In this paper we consider this problem. DEFINITION. A ring D is said to be totally integrally closed if for any ring homomorphism o: B-+D and any integral extension C of B there is a homomorphism C--D extending a. (The term integrally closed is reserved for an integral domain which is integrally closed in its field of fractions.) The following two propositions are immediate.

The Structure of QF-3 Rings

Abstract: Introduction. In '[12] Thrall introdcuced three generalizations of the quasiFrobenius (=QF) algebras of Nakayama [9], [10]. In this paper we shall be concerned with ring theoretic generalizations of two of these Thrall algebras-namely QF-2 algebras and QF-3 algebras. If R is a ring then a one-sided ideal of R is primitive in case it is generated by a primitive idempotent, and an R-module is minimalfaithful in case it is faithful and has no proper faithful direct summand. Extending Thrall's original definitions to (two-sided) artinian rings we have: QF-2 rings: An artinian ring is QF-2 in case each of its primitive one-sided ideals has a simple socle. QF-3 rings: An artinian ring is QF-3 in case it has (to within isomorphism) a unique minimal faithful left module. It is not difficult to show that QF rings are both QF-2 and QF-3 (see [2, ??58-59]). Moreover, Thrall [12] has shown that QF-2 algebras are QF-3 but not necessarily QF. Most of the information about QF-2 and QF-3 rings is limited to finite dimensional algebras (see [8], [12], [13]). Two notable exceptions generalize to QF-3 rings results known to hold for QF-3 algebras almost from their inception. Specifically, Jans [7] has characterized QF-3 rings as those artinian rings whose left injective hulls are projective and Harada [5] has shown that the QF-3 property is actually " two-sided " (i.e., a QF-3 ring has a unique minimal faithful right module). In ?2 of this paper we obtain ideal theoretic characterizations of the injective projective modules (and hence of the unique minimal faithful module) over "left QF-3 rings". Our main results appear in ?3. With the aid of Morita's duality theorems [8] we obtain characterizations of QF-3 rings that are analogous to Nakayama's original definition of QF rings in terms of socles of primitive one-sided ideals [10], his characterization of QF-rings in terms of the double annihilator property for one-sided ideals [10], and the fact (see [8, ?14]) that QF rings are precisely those artinian rings for which the functor HomR ( , R) provides a duality between the categories of finitely generated left and finitely generated right R-modules.

Generalized Radical Rings

Abstract: Let R be a ring. We denote by o the so-called circle composition on R, denned by a o b = a + b — ab for a, b ∊ R. It is well known that this composition is associative and that R is a radical ring in the sense of Jacobson (see 6) if and only if the semigroup (R, o) is a group. We shall say that R is a generalized radical ring if (R, o) is a union of groups. Such rings might equally appropriately be called generalized strongly regular rings, since every strongly regular ring satisfies this property (see Theorem A below). This definition was in fact partially motivated by the observation of Jiang Luh (7) that a ring is strongly regular if and only if its multiplicative semigroup is a union of groups.

On rings in which every ideal is the annihilator of an element

Abstract: Let R be a semisimple Artin ring (i.e. R is a direct sum of matrix rings over division rings). Then if L is a left ideal of R, L = Re where e is an idempotent and so L is exactly the left annihilator of the element 1 -e. We investigate the structure of rings having this property. If S is a subset of a ring R, let e(S) and r(S) denote the left and right annihilators of S. The notation eR(S) will be used when it is necessary to specify the ring R. DEFINITION. R is a left elemental annihilator ring (l.e.a.r.) if, whenever L is a left ideal of R, there exists an element aER such that L=e(a). A right elemental annihilator ring (r.e.a.r.) is defined analogously. Notice that if R is any ring (always assumed to have a unity element) and S is a subset of R, then e(S)=e(r(e(S))) and r(S) =r(e(r(S))). Hence if R is a l.e.a.r., and L is a left ideal, we have in particular that L==e(r(L)). If R is a r.e.a.r., and I a right ideal, then I=r(e(I)). We consider first the commutative case. It is known [1, Theorem 1.1] that if R is completely primary (local with nilpotent maximal ideal), then R has the property that every ideal is the annihilator of some subset of R if and only if R has a unique minimal ideal. Using the methods of Theorem I below, it is possible to derive from this the fact that if R is a commutative noetherian ring, then every ideal of R is the annihilator of some subset if and only if R is a direct sum of completely primary rings each of which has a unique minimal ideal. By imposing the more strenuous condition that R actually be an elemental annihilator ring, we obtain a similar result without the hypothesis of the chain condition.

The Jacobson Radical of Some Endomorphism Rings

Abstract: In what follows, G always denotes an abelian group where OG is its zero and EG is its endomorphism ring. If S is a ring, then JS is to be its Jacobson radical, Os is to be its zero, while its unity, if any, is denoted by 1s. In ansver to a query of Jacobson’s [3, p. 23], Patterson [5] [4] showed that the Jacobson radical of the ring of row-finite matrices over S is the ring of row-finite matrices over JS if and only if JS has a right-vanishing condition due to Levitzki, namely that if b0,b1,... is any sequence chosen from JS, then there is a least non-negative integer n (depending upon the sequence) for which the product bO...bn (just bO if n = O) vanishes. If n depends only upon the initial sequence member b, then [4] this sort of right vanishing is called uniform.

Semi-primary ${\rm QF}-3$ rings

Abstract: A ring R (with identity) is semi-primary if it contains a nilpotent ideal N with R/N semi-simple with minimum condition R is called a left QF-3 ring if it contains a faithful projective injective left ideal If R is semi-primary and left QF -3, then there is a faithful projective injective left ideal of R which is a direct summand of every faithful left R -module [5], in agreement with the definition of QF -3 algebra given by RM Thrall [6] Let Q ( M ) denote the injective envelope of a (left) R -module M We call R left QF -3 + if Q ( R ) is projective JP Jans showed that among rings with minimum condition on left ideals, the classes of QF -3 and QF -3 + rings coincide [5]

A Ring Structure Processor for a Small Computer

Abstract: A low-level data structure package for the PDP7 computer is described. Its principal features are the compact form in which given structures may be set up and the wide range of formats permitted. Ring structures are regarded as special cases of general list structures, and the package permits the generation and processing of all legal list structures. It is, however, specifically oriented to a certain class of uni-directional list and ring formats, and achieves particularly good space utilization when they are used. Space statistics for the package are presented which, when compared with the performance of more conventional schemes, show typical savings of about 30%.

A Theorem on Henselian Rings

Abstract: It is known that if K is a field, then the ring of formal power series in one or more variables, with coefficients in K, is Henselian at its maximal ideal. In this note we show that if R is a ring (commutative and with identity element) which is Henselian at the maximal ideals M1, M2, … then R[[x]] - the ring of formal power series in x with coefficients from R - is also Henselian at the maximal ideals M1 ⋅ R[[x]] + x⋅ R[[x]], etc.

On the construction of upper radical properties

Abstract: PROOF For each ring R, let DI(R) 1 I: I is an ideal of R } Assuming Dn(R) has been defined, let Dn+1(R) { I: I is an ideal of some ring in Dn(R) } and defined D(R) = U I Dn(R): n = 1, 2, * * * } Let 9YD be the class of all rings A such that A is isomorphic to some ring in D(R) for some R in W) If A is an element of 9), then there exists a positive integer n and a ring R an element of 9) such that A is isomorphic to a ring I in Dn(R) Therefore, if J is a nonzero ideal of A, then J is isomorphic and hence homomorphic to an ideal J' of I and J' EDn+?(R) and hence J' E9) and 9) satisfies the hypothesis of Theorem 1 Thus 9) determines an upper radical property (5g and every ring in 9)1 and in particular in 9) is 25-semisimple Next we want to show that 25p is the upper radical property determined by WI Let 25 be any radical property such that every ring in 9) is S-semisimple For each RE9), by Theorem 2, every ring in D1(R) is e-semisimple and by induction, every ring in D(R) is 2semisimple and hence every ring in 9) is 25-semisimple Since (5g is the upper radical property determined by 91, it follows that 25Ce

About linear groups over rings

Abstract: 1. The (~-Radical. We shall prove in this section the following theorem. THEOREM I. An exterior radical of the group r is a Z-group. LEMMA 1. ff K is a commutative ring and R is its radical, then there exists in R a system of ideals of the ring K, annihilating in R. We ay that a system {~B} of ideals of a ring K is annihila*ing in R if for any jump RBc Rfl+! the inclusions R/3+vR cR~, R-R~+~cRfl hold. m ca~,e of a commutative ring K one il)clusion is, obviously, sufficient. Let {R~ be a composition Sy.~tem of all- right ideals of the ring K contained in R. Since K is a commutative ring, Rfl are even twosided idea~s in K. We shall examine the composition p~" Rfl, Rfl+ 1, R~ c Rfl+ I. The factor ring Rfl+I/Rf~ canbe considered as a right K-module, moreover, since the pai. Rfl, R- - RB c Rfl+. is a composition pair, it is an irreducible module. Jacobson radical consists by definition .~+1, 1 [2] of such and only sdch elements of the ring K, which act as zeroa in every irreducible K-module. Therefore, the inclusion R-Rfl+ic Rfl holds, hence the. system {Bfl} is annihilating. Proof of Theorem 1. We shall denote by H the subalgebra of Kn generated by the set (~(F) -F. We shaIl show that the algebra H has an annihilating system. If J is a primitive ideal of the ring K, and JK n is the corresponding submodule (jgn consists of all rows with elements from the ideal J}, then the factor-module Kn/JK n is at. n-dimensional vector space over the field K/J. Let Hj be the kernel of the rePresentation (Kn/~rK n, H}. The algebra H acts by definition as a zero in every F-composition f~ or. Since the length of a r-composition series in Kn/JK n does not exceed n, it follows that the algebra H/Hj is a nilpotcnt algebra of rank -< n-1. Let {J~}a~A be the system of all primitive ideals of the ring K. Then N J(~ = R hence 0 Hj -- H N Rn. o~EA cr Remak's theorem implies now that the algebra H/H N R n is nilpotent, m will denote the degree of nilpotency of the algebra H/H N R n. The subalgebra H N R n has an annihilating system in virtue ef Lemma 1. Indeed, if {Rfl} is an annihilating system in R, then {(Rfl}n} is ae. annihilating system in R n (here (Rfl)n is an ideal of Kn, consisting of all matrices with elements from R/3}. The set {H n (Rfl}n} bec'~mes, ~ter the repetitions, if any, are deleted, an annihilating system in H N Rn already.

Variation of a Complex Structure in a Point

Abstract: We shall consider the following problem: given a topological space E, a point e0 of E, and a complex structure on E\{eo}, which can be extended to a complex structure on E, describe the totality of these complex structures on E. It will turn out that this set is, in a canonical way, a complex space, which we shall describe in an axiomatic way. Its connected components are projective algebraic. Choosing E as the open unit disc and the usual complex structure on E\{e0} we obtain a survey on all isomorphy classes of 1-dimensional irreducible germs of complex spaces. Of course the same can be done for reducible germs and for other cases. If X is a topological space and x E X we denote by Ex the algebra of germs at x of continuous functions defined in open neighborhoods ofi x. Let fl,x , f, be elements of the maximal ideal of 9. Then, for any convergent power series b in n variables, ( (fx,. , fn,) is an element of 9x defined in an obvious way, which we call a convergent power series in fl, f , fx. All convergent power series in f,x, * * * , f. form a subring of 9. This ring is denoted C[ ]. We use this ring to describe a complex structure in the following lemma.

The structure of the category of sheaves in the flat topology over certain local rings.

Abstract: Introduction. The local rings, A, of the totle are complete, neotherian, local rings of dimension 0, [3, Th. 2] proved useful in the theory of cohomological dimension. By characterizing the etale sheaves (see ? 1) over a special local ring (Th. 1), we are able to further generalize these structure theorems to obtain the canonical decomposition of a sheaf over such a ring into infinitesimal and etale parts, (Th. 2). Hopefully, our structure theorem will help in the investigation of cohomological phenomena over special local rings.

Hochschild cohomologies for z-rings with a power basis

Abstract: Let λ be an associative ring with unity. The main result of the article consists in the proof of the periodicity of the Hochschild cohomologies of λ in the case when λ is a Z-ring with a power basis. The period is equal to 2. This result is proved for maximal orders of fields of algebraic numbers.