Showing papers on "Ring (mathematics) published in 1971"

The Spectrum of an Equivariant Cohomology Ring: II

Abstract: Let G be a compact Lie group (e.g., a finite group) and let HG= H*(BG, Z/pZ) be its mod p cohomology ring. One knows this ring is finitely generated, hence upon dividing out by the ideal of nilpotent elements it becomes a finitely generated commutative algebra over the field Z/pZ. It is the purpose of this series of papers to relate the invariants attached to such a ring by commutative algebra to the family of elementary abelian p-subgroups of G. For example we prove a conjecture of Atiyah and Swan to the effect that the Krull dimension of the ring equals the maximum rank of an elementary abelian p-subgroup. Another result, which will appear in part II, asserts that the minimal prime ideals of the ring are in one-one correspondence with the conjugacy classes of maximal elementary abelian p-subgroups. Actually the theorems of the series are formulated more generally for the equivariant cohomology ring of a G-space X, defined by the formula

Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci

Abstract: 0. Introduction. Let R be a commutative Noetherian ring with identity and let I be a proper ideal of R. A classical result of Krull is that if I can be generated by r elements then the rank or altitude of I (the greatest rank of any minimal prime of I) is at most r. If, moreover, the grade of I (the length of the longest R-sequence contained in I) is r, then I enjoys certain special properties summarized in the term "perfect" as used by iRees [30, p. 32]: I is perfect if the homological (or projective) dimension of R/I as an R-module is equal to the grade of I. The associated primes of a perfect ideal I all have the same grade as I, that is, perfect ideals are grade unmixed. If R is Cohen-Macaulay, the grade of any ideal is equal to its little rank of height (the least rank of any minimal prime) ; in particular, the notions of grade and rank coincide on -primes, and perfect ideals are rank unmixed. Moreover, if I is perfect in a Cohen-Macaulay ring R, R/I is again (Cohen-Macaulay. Macaulay's famous theorem that in a polynomial ring over a field a rank r ideal which can be generated by r elements is rank unmixed [36, p. 203] is then a consequence of two facts: a polynomial ring over a field is CohenMacaulay, and a grade r ideal generated by r elements is perfect. This is the classical example of a perfect ideal. Good discussions of the subject. are available: see [9], [24, ? 25], [30], [18, Ch. 3], and [36, Appendix 6]. The Noetherian restriction on R is, for certain purposes, unnecessary in the discussion of perfect ideals, if one adopts a suitable definition of grade. This idea is worked out in [1]. Suppose that R is (locally) regular, and I is an ideal of R such that R/I is not the direct product of two rings in a nontrivial way. Then I is perfect if and only if R/I is Cohen-Macaulay. In particular, this is the situation when R is a polynomial ring over a field and I is homogeneous. It is very natural, then, to hunt for perfect ideals. Relatively few classes are known, but several authors [4, 6, 8, 33] have established the perfection

Thermodynamics of the Heisenberg-Ising Ring for Δ >~ 1

Abstract: The thermodynamics of the Heisenberg-Ising ring is reduced to the solution of a system of recurrent nonlinear integral equations.

Rings and modules of quotients

Abstract: The development of a general theory of rings of quotients may be said to have started with the construction by Johnson [1] of the maximal ring of quotients Qmax of a non-singular ring A. This was done before the theory of injective envelopes had become available, but it was later proved that Qmax could be used as an injective envelope of the ring A. The maximal ring of quotients of an arbitrary ring A was defined by Utumi [1] and studied by Findlay and Lambek [1]. In particular, Lambek [2] proved that it could be interpreted as the bicommutator of the injective envelope of A.

Formal groups, power systems and adams operators

Abstract: This paper provides a systematic presentation of the connection between the theory of one-dimensional formal groups and the theory of unitary cobordism. Two new algebraic concepts are introduced: formal power systems and two-valued formal groups. A presentation of the general theory of formal power systems is given, and it is shown that cobordism theory gives a nontrivial example of a system which is not a formal group. A two-valued formal group is constructed whose ring of coefficients is closely related to the bordism ring of a symplectic manifold. Finally, applications of formal groups and power systems are made to the theory of fixed points of periodic transformations of quasicomplex manifolds. Bibliography: 17 citations

Rings of definition of dense subgroups of semisimple linear groups

Abstract: We investigate the question: What is the smallest ring over which the elements of a dense subgroup (in the Zariski topology) of a semisimple algebraic group can be written down simultaneously for various rational linear representations?

The structure of serial rings.

Abstract: A serial ring (generalized uniserial in the terminology of Nakayama) is one whose left and right free modules are direct sums of modules with unique finite composition series (uniserial modules.) This paper presents a module-theoretic discussion of the structure of serial rings, and some onesided characterizations of certain kinds of serial rings. As an application of the structure theory, an easy proof is given of A. W. Goldie's characterization of serial rings with trivial singular ideal.

Extensions of derivations

Abstract: We show that for a class of algebras including separable algebras one can extend derivations of the center to derivations of the algebra. The following theorem was proved in the special cases that C is a field by Hochschild (Ho) and C is a semilocal ring by Roy and Sridharan (R, S) (and for any C in (Kn)). It is also a trivial conse- quence of a more general result proved by a short cohomological argument. Theorem 1. Let A be an algebra separable over its center C and M be an A®c A°r'-module. Then any derivation d '. C—>MA extends to a derivation d:A—*M. Since an algebra separable over its center C is C-projective (A, G,

Formal groups and their role in the apparatus of algebraic topology

Abstract: CONTENTSIntroduction § 1. Formal groups § 2. Cobordism and bordism theory § 3. The formal group of geometric cobordisms § 4. Two-valued formal groups and power systems § 5. Fixed points of periodic mappings in terms of formal groupsAppendix I. Steenrod powers in cobordisms and a new method of computing the bordism ring of quasicomplex manifoldsAppendix II. The Adams conjectureReferences

Shift register interconnection system

Abstract: Units of a data processing system communicate on a ring connection of shift register stages. The number of stages in a shift register is made small to avoid the delays that accompany the long data paths of a large ring system. Interconnecting stages are provided to direct a message on a first ring to a second ring according to an address contained in the message. Several useful configurations are disclosed. With this arrangement, a system of small rings can be expanded without correspondingly lengthening the average time for transmitting a message in the system.

Structure of witt rings, quotients of abelian group rings, and orderings of fields

Abstract: In particular, Theorems 5 and 6 yield the results of [5, §3] for Witt rings of formally real fields and Theorem 7 those of [5, §5] for Witt rings of nonreal fields. By studying subrings of the rings described in Theorems 5-7 and using the results of [2] for symmetric bilinear forms over a Dedekind ring

On functional representations of a ring without nilpotent elements

Abstract: In [3, p. 149], J. Lambek gives a proof of a theorem, essentially due to Grothendieck and Dieudonne, that if R is a commutative ring with 1 then R is isomorphic to the ring of global sections of a sheaf over the prime ideal space of R where a stalk of the sheaf is of the form R/0 P , for each prime ideal P, and . In this note we will show, this type of representation of a noncommutative ring is possible if the ring contains no nonzero nilpotent elements.

Dedekind subrings ofk[x 1,…,x n ] are rings of polynomials

Abstract: The purpose of this note is to prove that a Dedekind domain R which contains a field k, and which is a subring ofk[x 1,…,x n ] is a ring of polynomials. This generalizes similar results of A. Evyatar and A. Zaks on principal ideal domains, and of P. M. Cohn for the casen=1. Our methods and proofs differ from those introduced previously.

Direct product of division rings and a paper of Abian

Abstract: It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian. Introduction. Let R be an associative ring equipped with the binary relation (^) defined by xay if and only if xy = x2 in R. In this paper, it is shown that ( ^ ) is an order relation on R if and only if, R has no nilpotent elements i9*0). Conditions on the binary relation (g) in order that R split into a direct product of division rings are then studied in the light of Abian's result (l, Theorem l). Using similar argumentation and using certain subdirect representation of rings with no nilpotent elements, one obtains a complete similarity with the commutative case (yet, no extra complication in the computa- tions). Conventions. R is an associative ring which is, unless otherwise stated, with no nilpotent elements (other than 0). As a result of (2), R can be embedded into a direct product of skewdomains, R—* YLiei £i (that is to say, rings R, having no one-sided divisors of zero). The former embedding is fixed throughout the paper. It is therefore legiti- mate to identify any element x in R with the family consisting of all its projections (xj.e/. Finally, all definitions in (l) are extended (verbatim) to the present case (of a noncommutative ring R) and are freely used throughout.

Two remarks about hereditary orders

Abstract: In the first remark it is shown that, over a Dedekind ring, hereditary orders in a separable algebra are precisely the "maximal" orders under a relation stronger than inclusion (Theorem 1). At the same time simple proofs for known structure theorems of hereditary orders are obtained. In the second remark a complete classification is given of lattices over a hereditary order, provided the underlying Dedekind ring is contained in an algebraic number field and the lattices satisfy the Eichler condition (Theorem 2). Let o be a Dedekind ring with quotient field k, A/k a separable finite-dimensional algebra over k and R an o-order in A (i.e. a finitely generated o-algebra in A, containing the identity and such that kR =A). An order R is hereditary, if every left ideal is a projective R-module. It is a classical result-apart from terminology-that maximal orders are hereditary, but the converse of this is false: there are nonmaximal hereditary orders. Our first remark is, that if inclusion is replaced by a stronger relation, hereditary orders are characterized by the property of being locally maximal everywhere under this relation. To avoid confusion, we will use the term extremal orders instead. This characterization of hereditary orders can be used to give very simple proofs of some known properties of hereditary orders, which were obtained by Harada [4] and Brumer [2]. Since Brumer [2] is not available in print, we include proofs of the main results given there. In the complete local case, the structure of Rp-lattices is well known (Brumer [2 ]). The basic fact is that indecomposable Rp-lattices are in fact lattices over a maximal order containing Rp. This does not hold globally and only partial results are known in that case. Using results of an earlier paper (Jacobinski [5]), we give a complete classification of lattices over a hereditary o-order, provided the quotient field of 0 is an algebraic number field. The local theory yields a classification of genera of R-lattices. Our result is that the lattices in a restricted genus are isomorphic. This means that two R-lattices M and N are Received by the editors April 24, 1970. AMS 1970 subject classifications. Primary 16A18, 16A14; Secondary 20C0, 16A50.

A class of balanced non-uniserial rings

Abstract: Let R be a ring with unity. An R-module M is called balanced, if the natural homomorphism from R to the double centralizer of M is surjective. If every left R-module is balanced, R is said to be left balanced (or to satisfy the double centralizer condition for left modules). It is well-known that every artinian uniserial ring is both left and right balanced, and recently Jans [3] conjectured that "if R has minimum condition, then every R-module has the double centralizer condition if and only if R is a uniserial ring". This conjecture has been proved in [-1] to be true for rings which are finitely generated over their centres. However, the following theorem shows that, in general, the conjecture is false.

A Demonstration of Lenz' Law?

Abstract: The generally accepted explanation for the jumping ring demonstration is reviewed and shown to be inadequate to describe the observed phenomena. A more complete explanation is brought to the readers' attention in this paper.

On one sided ideals of a prime type

Abstract: The notion of prime type for right ideals is analogous to that of prime ideal in a commutative ring, and the right dimension of the ring R is defined using chains of right ideals of prime type. The meaning of zero right dimension, finite dimension for right primitive rings, and related topics are studied.

Subgroups of general and special linear groups over a ring

Abstract: A description of parabolic subgroups of the group SL(n, R) over a local ring. Some intermediate subgroups are described for the case of a euclidean ring. It is noted that analogous results hold for GL(n, R).

Rotating relativistic ring.

Abstract: Consideration of a rotating ring sheds light upon the problem of the rotating disk recently discussed in Nature by several authors and shows the disk problem to be more difficult than it appears.

A weak Nullstellensatz for valuations

Abstract: Given a real-valued pseudovaluation p on a commutative ring R, we show how to obtain a valuation v greater than or equal to p, and also satisfying certain upper bounds: in particular, if p(st) =p(s) +p(t) for all s, tES, S a multiplicative semigroup in R, then v can be chosen so that v(s) =p(s) for all sES. 1. An important lemma of commutative ring theory-a form of the "weak Nullstellensatz"-says that given an ideal I of a ring R, and a multiplicative semigroup S in R disjoint from I, there exists a prime ideal p containing I, and still disjoint from S. Now a pseudovaluation on a ring can be considered analogous to an ideal-an ideal tells us which elements to "consider 0", a pseudovaluation tells us which elements to "consider small". In particular, the valuations are like the prime ideals. We shall prove here some analogs to the lemma quoted above, showing how to obtain valuations from pseudovaluations. The desire to give the strongest possible result has made the statement of our first theorem (and the two lemmas used to prove it) rather complicated. But the special cases that follow it are more modest, and more handleable. The analogy with ideal theory can be made precise by noting that there is a 1-1 correspondence between ideals in R, and pseudovaluations of R into the two-element additive semigroup {O, + 0o }. Each of the results proved below, if stated for this semigroup rather than RU { + 0o }, is a result about ideals; and these statements follow from those proved via the observation that { 0, +?o0 } is a retract of RU { + oo }, as ordered semigroups. 2. Let R be a commutative associative ring with unit. We will designate by P the additive ordered semigroup of real numbers with + oo adjoined. DEFINITIONS. By a pseudovaluation on R, we shall mean a function p: R->P satisfying: (1) p(l) = O, p(O) -+ o, Received by the editors February 10, 1970. AMS 1970 subject classifications. Primary 13A15; Secondary 12J20, 14A05.