Showing papers on "Ring (mathematics) published in 1981"

The evolution of an elliptic vortex ring

Abstract: The evolution of a vortex ring in an ideal fluid under self-induction from a flat and elliptic configuration is followed numerically using the cut-off approximation (Crow 1970) for the velocity at the vortex. Calculations are presented for four different axes ratios of the initial ellipse. A particular choice is made for the core size and vorticity distribution in the core of the vortex ring. When the initial axes ratio is close to 1, the vortex ring oscillates periodically. The periodicity is lost as more eccentric cases are considered. For initial axes ratio 0·2, the calculations suggest a break-up of the ring through the core at one portion of the ring touching that at another, initially distant, portion of the ring.Results from quantitative experiments, conducted at moderate Reynolds number with the vortex rings produced by puffing air through elliptic orifices, are compared with the calculations. The agreement is fairly good and it is found that a vortex ring produced from an orifice of axes ratio 0·2 breaks up into two smaller rings. The relevance of the results to the vortex trail of an aircraft is discussed.

Symmetry breaking in distributive networks

Abstract: Given a ring (cycle) of n processes it is required to design the processes so that they will be able to choose a leader (a uniquely designated process) by sending messages along the ring. If the processes are indistiguishable there is no deterministic algorithm, and therefore probabilistic algorithms are proposed. These algorithms need not terminate, but their expected complexity (time or number of bits of communication) is bounded by a function of n. If the processes work asynchronously then on the average O(n log2n) bits are transmitted. In the above cases the size n of the ring was assumed to be known. If n is not known it is suggested first to determine the value of n and then use the above algorithm. However, n may only be determined probabilistically and any algorithm may yield an incorrect value. In addition, it is shown that the size of the ring cannot be calculated by any probabilistic algorithm in which the processes can sense termination.

Almost distributive lattices

Abstract: The concept of ‘Almost Distributive Lattices’ (ADL) is introduced. This class of ADLs includes almost all the existing ring theoretic generalisations of a Boolean ring (algebra) like regular rings, P-rings, biregular rings, associate rings, P1-rings, triple systems, etc. This class also includes the class of Baer-Stone semigroups. A one-to-one correspondence is exhibited between the class of relatively complemented ADLs and the class of Almost Boolean Rings analogous to the well-known Stone's correspondence. Many concepts in distributive lattices can be extended to the class of ADLs through its principal ideals which from a distributive lattice with 0. Subdirect and Sheaf representations of an ADL are obtained.

Ring polymers in solution : topological effects

Abstract: The effect of topological constraints on the properties of ring polymers in solution are studied. When the rings are short and rigid, the effects can easily be understood and a simple result is given here. When the rings are long and flexible, the situation is complex and a more subtle analysis is needed. Fortunately recent mathematical studies concerning the linking numbers of two curves lead to a significant result. This information is used to argue that the topological constraints produce essentially an increase of the local excluded volume interaction ; this topological effect could therefore be taken into account within the framework of current theories.

Derivations with nilpotent values

Abstract: IfR is a semiprme ring andd a derivation ofR such thatd(x) n=0 for allx∈R, wheren≥1 is a fixed integer, thend=0.

On spectrum placement for linear time invariant delay systems

Abstract: A ring of delay operators is used to obtain a ring model description of a linear time invariant delay system. With this aid an algorithm is developed for the computation of the polynomial state-feedback matrix which assigns an arbitrary set of poles of the closed-loop system (spectrum placement). An example is included to illustrate the method.

Rings associated with ideals and analytic spread

Abstract: Let A be a Noether ring and let = ( a 1 ,…, a r ) be an ideal of A . There are a number of graded rings that we can associate with . In this paper we shall be concerned with the following.

Baer rings and quasicontinuous rings have a MDSN

Abstract: The notion of a direct summand of a ring containing the set of nilpotents in some "dense" way has been considered by Y. Utumi, L. Jeremy, C. Faith, and G. F, Birkenmeier. Several types of rings including right selfinjective rings, commutative FPF rings, and rings which are a direct sum of indecomposable right ideals have been shown to have a MDSN (i.e., the minimal direct summand containing the nilpotent elements). In this paper, the class of rings which have a MDSN is enlarged to include quasiBaer rings and right quasi-continuous rings. Also, several known results are generalized. Specifically, the following results are proved: (Theorem 3) Let R be a ring in which each right annihilator of a reduced (i.e., no nonzero nilpotent elements) right ideal is essential in an idempotent generated right ideal. Then R^A®B where B is the MDSN and an essential extension of Nt (i.e., the ideal generated by the nilpotent elements of index two), and A is a reduced right ideal of R which is also an abelian Baer ring. (Corollary 6) Let R be an ATF*-aIgebra. Then R = A®B where A is a commutative AW*-algebra, and B is the MDSN of R and B is an ATF*-algebra which is a rational extension of Nt. Furthermore, A contains all reduced ideals of R. (Theorem 12) Let R be a ring such that each reduced right ideal is essential in an idempotent generated right ideal. Then R = A 0 B where B is the densely nil MDSN, and A is both a reduced quasi-continuous right ideal of R and a right quasi-continuous abelian Baer ring.

Why a ring

Abstract: In a world increasingly populated with Ethernets and Ethernet-like nets a few sites continue to experiment with rings of active repeaters for local data communication. This paper explores some of the engineering problems involved in designing a ring that has no central control, and then compared the M.I.T.-designed ring with Ethernet on a variety of operational and subtle technical grounds, on each of which the ring may possess important or interesting advantages.

Topological distribution functions and the second virial coefficients of ring polymers

Abstract: A new method (a topological moment method) is presented for treating statistical problems of ring polymer chains of which the topological state (Gauss linking coefficient) is prescribed. It is shown that the topological interaction among ring polymers formally resembles the excluded volume effect among linear polymers. A ’’topological parameter γ’’ which corresponds to the binary cluster integral β in the excluded volume problem is newly introduced for representing the strength of the topological interaction among submolecules. The parameter γ is given as a function of bond length, bond angle, internal‐rotational potential, and of interaction potentials among chain elements. Analytical expressions for the distribution function of the intermolecular distance and the second virial coefficient are derived and they are computed numerically.

Graham Higman's Thesis “Units in Group Rings”

Abstract: pp. 1-6) In the Abstract, Higman describes his results. His interests and emphases are clearly expressed in it and for this reason I include it in its entirety. It also defines his notation. In my commentary I use a form of notation more common at present. Having parallel notations is clumsy at times, mainly for tings of coefficients. Higman uses K for a general ring or field of coefficients; I have used R for a ring and K for a field. He uses C for a ring of algebraic integers; I have again used R with K being the corresponding quotient field (Higman uses k). My notation for the augmentation ideal of the group ring RG is I(G), with I(H)RG denoting the ideal of RG generated by I(H), H a normal subgroup of G; Higman uses

Lying-Over Pairs of Commutative Rings

Abstract: (R, T) is said to be a lying-over pair in case R ⊂ T is an extension of (commutative) rings each of whose intermediate extensions possesses the lying-over property. This paper treats several types of extensions, including lying-over pairs, which figure in some known characterizations of integrality. Several new characterizations of integrality are thereby obtained; as well, our earlier characterization of P-extensions is sharpened with the aid of a suitable weakening of the incomparability property. In numerous cases, a lying-over pair (R, T) must be an integral extension (for example, if R is quasisemilocal or if (R, T) is a coherent pair of overrings). However, any algebraically closed field F of positive characteristic has an infinitely-generated algebra T such that (F, T) is a lying-over pair. For any ring R, (R, R[X]) is a lying-over pair if and only if R has Krull dimension 0. An algebra T over a field F produces a lying-over pair (F, T) if and only if T is integral over each nonfield between F and T.

Semiprime rings with nilpotent derivatives

Abstract: There has been a great deal of work recently concerning the relationship between the commutativity of a ring JR and the existence of certain specified derivations of R. Bell, Herstein, Procesei, Schacher, Ligh, Martindale, Putcha, Wilson, and Yaqub [1, 2, 6, 8, 9, 10, 11, 12, 14] have studied conditions on commutators which imply the commutativity of rings.

Projective modules over rings with many units

Abstract: Let R be a commutative ring. Assume that every polynomial whose values generate the unit ideal actually takes on an invertible value. Then projective R-modules split into cyclic summands, and those of constant rank are free. A ring R (commutative with 1) satisfies the primitive condition if each f(x) = ao + + anXn that is primitive (Y:(aiR) = R) has some b in R with f(b) a unit. This condition, which guarantees the existence of many units in R, was introduced by van der Kallen [13]; he gave examples of rings satisfying the condition and established properties of K2(R) for such R. Subsequently it was shown [4], [5], [6] that the condition implies pleasant structural results about GL2(R) and Aut(GL2(R)). One step in this was a computational argument [6, 11.3] proving that if Q is a rank one direct summand of F where F is free of rank 2, then Q is free. We here will see that a much more general result is true. It actually applies to a slightly larger class of rings, and we begin by discussing them. I. Let f(X1, ... , Xn) be a polynomial over a ring R. We will say that f has local unit values if for each maximal ideal M of R there are b,, . . ., bn in RM with f(bl, ... , bn) invertible in RM. We can here replace the bi by elements of R congruent to them modulo M, so the condition says that not all values of f are in M; in other words, the values of f should generate the unit ideal of R. (This implies, of course, that the coefficients of f must generate the unit ideal.) We say that f has unit values if somef(b1, . . ., bn) is actually invertible in R. The rings we care about will be those in which every f with local unit values has unit values. Since elements are invertible iff they are so modulo the Jacobson radical J(R), it is evident that R has this property iff R/J(R) does. In particular, semilocal rings have the property. It is also easy to see that a product HIRi has the property iff all the factors do (consider maximal ideals of the form Mi x fljH i Rj). Further examples of rings with this property are given by the following propositions. PROPOSITION. Let R be a ring for which R/J(R) is von Neumann regular (= absolutely flat). Then polynomials with local unit values have unit values. PROOF. Replacing R by R/J(R), we may assume it is von Neumann regular. Suppose f has local unit values. For each maximal M pick b = (bl, ... , bn) with Received by the editors August 28, 1980. 1980 Mathematics Subject Classification Primary 13CO5. 'The work of both authors was supported in part by the National Science Foundation. ? 1981 American Mathematical Society 0002-9939/8 1/0000-0503/$02.00 455 This content downloaded from 157.55.39.231 on Thu, 06 Oct 2016 04:38:41 UTC All use subject to http://about.jstor.org/terms 456 B. I{. McDONALD AND W. C. WATEIRHOUSE f(b) a unit at M; then f(b) is still a unit on a neighborhood of M in Spec R. Since Spec R is a Boolean space, we can refine this covering to a finite covering by disjoint clopen sets U where we have f(bu) invertible on U. We may then choose b agreeing with bu on U, and f(b) will be invertible. E] This result could also be deduced from [2, Proposition 2]. The argument shows more generally that if we have a sheaf of rings over a Boolean space and the fibers have our property, so does the ring of global sections. We should also point out that by [1, 11.4, Exercise 16, p. 173] we have the following special case of the proposition: COROLLARY. Let R be zero-dimensional. Then polynomials with local unit values have unit values. LI PROPOSITION. Let S be an R-algebra which is a finitely generated free R-module. Suppose that over R, all polynomials with local unit values have unit values. Then the same is true over S. PROOF. Let s1, ... , sm be a basis of S over R. Given f(X1, ... , XJ) over S, take indeterminates Y 11..., Ymn, and define a polynomial g( Y) over R as the norm (from S to R) of f(> si Yi, . .. , E si Yi). Then setting Xj = E sirij makes f(X) invertible iff g(rij) is invertible, since units and only units have unit norms. Assume now that f has local unit values. If M is any maximal ideal of R, then SM is semilocal, so f has unit values in SM. Hence g has unit values in RM. By hypothesis then g has unit values in R. C] The main theorem will automatically allow us to replace "free" by "projective of constant rank" in this result. We now show exactly how our property is related to the primitive condition mentioned in the introduction. For this we need a pair of simple lemmas. LEMMA. Let R satisfy the primitive condition. Let fi(X) = E aXJ be a finite sequence of polynomials with Eij(aijR) = R. Then there is some b in R with E(fi(b)R) = R. PROOF. Choose an integer m greater than the degrees of all fi, and let g(X) = L f(X)Xm'. All aij occur as coefficients in g, so g is primitive. Hence some g(b) = E f(b)b'm is a unit, and in particular 2(fi(b))R = R. E] This allows us to deduce a multivariable extension of the condition: LEMMA. Let R satisfy the primitive condition. Let f(XI,... , XJ) = E a,Xa be a polynomial with E(a,R) = R. Then there are bl, . .. , b, in R with f(b1, .. ., bn) invertible. PROOF. Rewrite f as E, f,(X1)X8, where ,B = (i2, . .. , in). All the a, appear as the coefficients of the polynomials ff(XI). By the lemma there is some b, such that E(ffi(bj)R) = R. Then f(bl, X2,... Xn) again satisfies the hypothesis of the lemma, and the result follows by induction. El PROPOSITION. A ring R satisfies the primitive condition iff (1) every polynomial with local unit values has unit values and (2) every residue field R/ M is infinite. This content downloaded from 157.55.39.231 on Thu, 06 Oct 2016 04:38:41 UTC All use subject to http://about.jstor.org/terms PROJECTIVE MODULES OVER RINGS 457 PROOF. If R/M has finite cardinality q, then Xq X is a primitive polynomial with all values in M. Thus if R satisfies the primitive condition, (2) must hold. And any f(XI, .. ., X",) with local unit values has coefficients generating the unit ideal, so (1) holds by the last lemma. Conversely, if f is a primitive polynomial, then it is nontrivial modulo M, so by (2) it has a nonzero value modulo M. This means it has local unit values, so by (1) it has unit values. E1 This shows in particular that rings with our property can be very far from dimension zero. Indeed [3], if A is any ring and S is the set of primitive polynomials in A[x], then R = S-'A[x] satisfies the primitive condition and has maximal ideal space identical with that of A.

Schur multiplier of a group of elementary matrices of finite order

Abstract: If a ring Λ is finitely generated as a module over its center, then for the Schur multiplier coincides with . For a representation of is obtained which is analogous to van der Kallen's representation for a commutative ring.

$X$-inner automorphisms of filtered algebras

Abstract: We continue earlier work and compute the X-inner automorphisms of the ring of differential polynomials in one variable over an arbitrary domain. This is then applied to iterated Ore extensions. We also show that the ring of generic matrices has no nonidentity automorphisms which fix the center. In this paper we continue the study of X-inner automorphisms of filtered algebras begun in [7], which will be referred to as (I) in what follows. In (I) we described the X-inner automorphisms of the enveloping algebra of a Lie algebra, and of the ring of differential polynomials A = R[x; d] in one variable over a commutative domain R. Here we first show that the ring of generic matrices has only the trivial X-inner automorphism; equivalently, any automorphism fixing the center is the identity automorphism. We then extend our result from (I) on differential polynomials, allowing an arbitrary prime ring R as the coefficient ring. When R is a domain, we explicitly determine the group of X-inner automorphisms of A = R[x; d] in terms of R and d. We then apply these results to certain iterated Ore extensions; in particular, if the original coefficient ring is commutative, then the group of all X-inner automorphisms is abelian. In what follows, A will always denote a prime ring. Recall that a E Aut(A) is X-inner if it becomes inner when extended to the (right) Martindale quotient ring QO(A) of A; when A is an Ore domain, a is X-inner if and only if it becomes inner on the quotient division ring D of A. We shall need the following properties of QO(A): it is the (right) quotient ring of A with respect to the filter IF of all nonzero two-sided ideals (that is, QO(A) = lim ,HomA('A, A)), it is a prime ring with center C C(A) a field, called the extended center of A, and A may be imbedded in QO(A) as left multiplications. By construction, for any 0 #d x E A, there exists a nonzero I of A so that 0 =# xI c A. Fundamental in what follows is an internal characterization of X-inner automorphisms [8]. LEMMA 1. If a E Aut(A) is X-inner, say r' = s-'rs, all r E A for some s E QO(A), then there exist nonzero 'a, b E A such that sa = b and arb = b0r0a, all r E A. Conversely if arb = b0r0a, 'all r E A, for some nonzero a, b E A, then there exists s E QO(A) with sa b which induces a. Received by the editors April 23, 1982 and, in revised form, July 19, 1982. 1980 Mathematics Subject Classification. Primary 16A72, 16A38. ' Research supported in part by NSF grant No. MCS 81-01730. ? 1983 American Mathematical Society 0002-9939/82/0000-0756/$02.75

Realization algorithms for systems over a principal ideal domain

Abstract: In this paper realization algorithms for systems over a principal ideal domain are described This is done using the Smith form or a modified Hermite form for matrices over a principal ideal domain It is shown that Ho's algorithm and an algorithm due to Zeiger can be generalized to the ring case Also a recursive realization algorithm, including some results concerning the partial realization problem, is presented Applications to systems over the integers, delay differential systems and two-dimensional systems are discussed

One theorem of Cohn

Abstract: Let F be the field of algebraic functions of one variable over the field of constants k, v be a point of field F/k, and Av be the ring of functions not having poles outside point v. It is proved that Av is a GE2-ring if and only if it coincides with the ring k[X] of polynomials of one variable over field k.