On group automata over finite fields

Associative rings R and R′ are said to be lattice-isomorphic if their subring lattices L(R) and L(R′) are isomorphic. An isomorphism of the lattice L(R) onto the lattice L(R′) is called a projection (or else a lattice isomorphism) of the ring R onto the ring R′. A ring R′ is called the projective image of a ring R. Lattice isomorphisms of finite one-generated rings with identity are studied. We elucidate the general structure of finite one-generated rings with identity and also give necessary and sufficient conditions for a finite ring decomposable into a direct sum of Galois rings to be generated by one element. Conditions are found under which the projective image of a ring decomposable into a direct sum of finite fields is a one-generated ring. We look at lattice isomorphisms of one-generated rings decomposable into direct sums of Galois rings of different types. Three main types of Galois rings are distinguished: finite fields, rings generated by idempotents, and rings of the form GR(pn,m), where m > 1 and n > 1. We specify sufficient conditions for the projective image of a one-generated ring decomposable into a sum of Galois rings and a nil ideal to be generated by one element.

The following theorem is proved: Suppose R is an associative ring and suppose J is the Jacobson radical of R. Suppose that for all x1, * *, xn in R, there exists a word wx... (x1, * * x), depending on x1, * *, x,,, in which at least one xi (i varies) is missing, and such that xi ... x, = w ,... .n(x1, , x). Then J is a nil ring of bounded index and R/J is finite. It is further proved that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent. In this paper, we investigate the structure of an associative ring R with the property that, for all xl, * * * , x, in R, xl ... x = wxi.... xn(xl, * * *, Xn), where w is a word, depending on xl, * * *, xn, and where some xi (i varies) is missing in w. For such a ring R, we prove the Jacobson radical J is a nil ring of bounded index. We also show that R/J is finite. Finally, we show that a commutative nil semigroup satisfies the above identity if and only if it is nilpotent. In preparation for the proofs of the main results, we first introduce DEFINITION 1. Let n be a positive integer, and let S be a semigroup. We say that S is a fln-semigroup if, for all xl, * * *, xn in S, there exists a word w .(x, *1 , xn) consisting of a product of the xi's with some x1 (j varies) missing, such that xl ... xn=wxl....xn((xl, * * *, xn). A ring R is called a fln-ring if its multiplicative semigroup is a fln-semigroup. LEMMA 1. Let S be a finite semigroup or a nilpotent semigroup. Then S is a fln-semigroup, for some n. PROOF. First, suppose S is finite, of order n, and suppose xl, Xn+1 are any elements of S. Then xi=xj for some i>j, and hence X1 ... Xn+1 = X1 ... Xi ... Xi-lXjXi+l Xn+1 = W(X1, . . , Xi-1 Xi+1 , Xn+1) Thus S is a fln+l-semigroup. Next, suppose S is nilpotent, say, Sm= (0). Presented to the Society, January 18, 1972; received by the editors August 10, 1971 and, in revised form, October 6, 1971. AMS (MOS) subject classifications (1970). Primary 16A38, 16A48; Secondary 20M10. The author was supported by a National Science Foundation Graduate Fellowship. @ American Mathematical Society 1973

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

For a commutative integral domain [Formula: see text] with field of fractions [Formula: see text], the ring of integer-valued polynomials on [Formula: see text] is [Formula: see text]. In this paper, we extend this construction to skew polynomial rings. Given an automorphism [Formula: see text] of [Formula: see text], the skew polynomial ring [Formula: see text] consists of polynomials with coefficients from [Formula: see text], and with multiplication given by [Formula: see text] for all [Formula: see text]. We define [Formula: see text], which is the set of integer-valued skew polynomials on [Formula: see text]. When [Formula: see text] is not the identity, [Formula: see text] is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that [Formula: see text] has a ring structure in many cases. We show how to produce elements of [Formula: see text] and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where [Formula: see text] is a discrete valuation ring with finite residue field.

We consider q-deformations of Witt rings, based on geometric operations on zeta functions of motives over finite fields, and we use these deformations to construct q-analogs of the Bost-Connes quantum statistical mechanical system. We show that the q-deformations obtained in this way can be related to Habiro ring constructions of analytic functions over F_1 and to categorifications of Bost-Connes systems.

Let M be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units G. The multiplicative semigroup .4 (F) , where F is a finite field, is a particular example. Using HarishChandra's theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of M is completely reducible. Using this we characterize the representations of G extending to irreducible representations of M as being those induced from the irreducible representations of certain parabolic subgroups of G. We go on to show that if F is any field and S any multiplicative subsemigroup of .4 (F) , then the semigroup algebra of S over any field of characteristic zero has nilpotent Jacobson radical. If S = .4 (F) , then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of .4 (F) over a field of characteristic zero is regular if and only if ch(F) = p > 0 and F is algebraic over its prime field.

We present a practical algorithm to construct the subgroup of the general linear group that preserves a system of bilinear or sesquilinear forms on a vector space defined over a finite field. Components include efficient algorithms to construct the Jacobson radical and the group of units of a matrix algebra.

Weightless cohomology of algebraic varieties

A group is called a ring if the group is a commutative under addition operation and satisfy the distributive and assosiative properties under multiplication operation. Suppose R is a commutative ring with non-zero identity, U is the unit of R, and J(R) is a Jacobson radical. Jacobson graph of a ring R denoted by ℑ R is a graph with a vertex set is R\\J(R) dan edge set is {(a, b)| 1 − ab ∉ U}. The purpose of this research is to construct a Jacobson graph of ring Z 3 n with n > 1. The results show that Jacobson graph of ring Z 3 n is a disconected graph with two components.

In her paper [3], Osofsky exhibited an example of a ring R containing 16 elements which (i) is equal to its left complete ring of quotients, (ii) is not self-injective and (iii) whose injective hull HR = H(RR) allows a ring structure extending the R-module structure of HR. In the present note, we offer a general method of constructing such rings; in particular, given a non-trivial split Frobenius algebra A and a natural n ≧ 2, a certain ring of n x n matrices over A provides such an example. Here, taking for A the semi-direct extension of Z/2Z by itself and n = 2, one gets the example of Osofsky. Thus, our approach answers her question on finding a non-computational method for proving the existence of such rings.

Secure multiparty computation refers to the problem where a number of users wish to securely compute a function on their inputs without revealing any unnecessary information. This dissertation focuses on the fundamental limits of secure summation under different constraints. We first focus on the minimal model of secure computation, in which two users each hold an input and wish to securely compute a function of their inputs at the server. We propose a novel scheme base on the algebraic structure of finite field and modulo ring of integers. Then we extend the minimal model of secure computation, in which K users wish to securely compute the sum of their inputs at the server. We prove a folklore result on the limits of communication cost and randomness cost. Then we characterized the optimal communication cost with user dropouts constraint, when some users may lose connection to the server and the server wishes to compute the sum of remaining inputs. Next, we characterize the optimal communication and randomness cost for symmetric groupwise keys and find the feasibility condition for arbitrary groupwise keys. Last, we study the secure summation with user selection, such that the server may select any subset of users to compute the sum of their inputs. This leads us to the MDS variable generation problem. We characterize the optimal individual key rate and the result is interestingly the harmonic number.

The Early Paleozoic collisional system located in the Olkhon region at the western shores of Lake Baikal resulted from collision of the Siberian paleocontinent and a complex aggregate composed by fragments of a microcontinent, island arcs, back-arc structures and accretionary prisms. The main events were associated with complete manifestation of shear tectogenesis initiated by oblique collision. The current structure includes tectonically displaced components of ancient geodynamic systems that used to have been located dozens and hundreds of kilometres apart. Horizontal amplitudes of tectonic displacement seem to have been quite high; however, numerical data are still lacking to support this conclusion. Information about the structure of the upper crust in the Paleozoic is also lacking as only deep metamorphic rocks (varying from epidote-amphibolite to granulite facies) are currently outcropped. Formations comprising the collisional collage are significantly different in composition and protoliths, and combinations of numerous shifted beds give evidence of a 'bulldozer' effect caused by the collisional shock followed by movements of crushed components of the ocean-continent zone along the margin of the Siberian paleocontinent. As evidenced by the recent cross-section, deep horizons of the Early Paleozoic crust comprise the collisional system between the Siberian craton and the Olkhon composite terrain. A permanent inclusion in the collisional combinations of rocks are unusual synmetamorphic injected bodies of carbonate rocks. Such rocks comprise two groups, marble melanges and crustal carbonate melted rocks. Obviously, carbonate rocks (that composed the original layers and horizons of stratified beds) can become less viscous to a certain degree at some locations during the process of oblique collision and acquire unusual properties and can thus intrude into the surrounding rocks of silicate composition. Such carbonate rocks behave as protrusions or intrusions and contain inclusions of silicate rocks. Formation of marble melanges is a multi-staged process: they occur as early tectonic covers and, more often, accompany shear zones of large lengths, comprise late push-out nappes initiated by shear faults, participate in construction of ring and vortex structures that are generated by shearing in the geological medium of inhomogeneous rheology. In general, the available data give evidence of the fact that formation of synmetamorphic marble melanges is a direct consequence of the oblique collision geodynamics and a sensitive indicator of such a regime. A pure guesswork may suggest that the occurrence of the marble melanges can be associated with a catastrophic loss of viscosity of the carbonate rocks due to a sharp increase of velocities of shear deformations that accompanied the oblique collision.

Detailed faunal analyses were conducted on two major subsistence resources, oysters and marine catfish, at Rollins Shell Ring, a Late Archaic (5000 – 3000 B.P.) site on the northeast coast of Florida. The focus of this investigation was on resource exploitation, and, specifically, whether there was evidence that oysters from this site were over-exploited. Three units from previous excavations at the site were selected for analysis, and represent the span of occupation recorded for this site. Measurements were obtained from oysters to determine habitat, and, along with shell height, were compared across time for any changes in the pattern of exploitation that would indicate over-exploitation. Marine catfish otoliths were used to provide seasonal data for oyster harvesting, as well as information on fish ages and harvesting of this resource. My analysis revealed that the majority of oysters used in the construction of the main shell ring, ringlets, and other structures at this site were harvested from the same habitat. While there were differences in oyster habitat exploitation and shell height between samples, the difference was attributed to the variability of oyster habitats exploited and shell height in the earliest sample of the analysis, Test Unit 10, and in the latest sample, Test Unit 2; there was less variability noted in oyster habitat and shell height for the middle activity period recorded at the site, Test Unit 12c. Further results indicated a seasonal preference for exploitation of both oyster and marine catfish in warm water temperatures, and that oyster resources did not appear to be under stress during the period of activity recorded for the site. These data suggest that it is unlikely that over-exploitation of oysters played a role in permanent site abandonment.

On monomial codes in modular group algebras

Difference families from rings