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The main aim of this paper is to investigate the relationship between the characteristic of a ring and the order of an element in the intuitionistic fuzzy environment. We also characterise the term divisor of zero in a ring and study their connection with the ideal of the ring in the intuitionistic fuzzy setting. Some related results are also analyse.

Let H be a finitely generated group of matrices over a field F of characteristic zero. We consider the group ring KH of H over an arbitrary field K whose characteristic is either zero or greater than some number N = N ( H ) . We prove that KH is isomorphic to a subring of a ring S which is a crossed product of a division ring Δ with a finite group. Hence KH is isomorphic to a subring of a matrix ring over a skew field.

Let R be a ring with identity. Then {0} and R are the only additive subgroups of R if and only if R is isomorphic (as a ring with identity) to (exactly) one of {0}, Z/pZ for a prime number p. Also, each additive subgroup of R is a one-sided ideal of R if and only if R is isomorphic to (exactly) one of {0}, Z, Z/nZ for an integer n ≥ 2. This note could find classroom use in a first course on abstract algebra as enrichment material for the unit on ring theory.

The article deals with the problems of mutual relations between Siberian local state bodies system and public peasantry authorities’ places. In the a set of laws called «Sibirskoe Uchrezhdenie» the supreme authorities set up methods and forms of work relations between bodies and places and gave order of mutual correspondence. The author characterizes forms of mutual correspondence, based on archive documents. The author proves with examples that public peasantry authorities addressed local state bodies only in accordance with the formal procedure prescribed for them by the supreme authorities.

In this paper we prove the following result. Let H be a real or complex Hilbert space, let L(H) be the algebra of all bounded linear operators on H and let A(H) ⊆ L(H) be a standard operator algebra. Suppose we have an additive mapping D:A(H) → L(H) satisfying the relation D(An)=D(A)A* n-1+AD(An-2)A* +An-1D(A) for all A A(H) and some fixed integer n>1. In this case there exists a unique B L(H) such that D(A)=BA*-AB holds for all A A(H).

This paper introduces the notion of zero ring labeling of a graph and its empirical study demonstrates that every graph admits a zero ring labeling with respect to some zero ring. The zero ring graph Γ ( R 0 ) turns out to be maximal with respect to an injective zero ring labeling. In particular, we determine the optimal zero ring index for some well-known graphs.

This work describes the ongoing specification and formalization in the PVS proof assistant of some definitions and theorems of ring theory in abstract algebra, and briefly presents some of the results intended to be formalized. So far, some important theorems from ring theory were specified and formally proved, like the First Isomorphism Theorem, the Binomial Theorem and the lemma establishing that every finite integral domain with cardinality greater than one is a field. The goal of the project in progress is to specify and formalize in PVS the main theorems from ring theory presented in undergraduate textbooks of abstract algebra, but in the short term the authors intended to formalize: (i) the Second and the Third Isomorphism Theorems for rings; (ii) the primality of the characteristic of a ring without zero divisors; (iii) definitions of prime and maximal ideals and theorems related with those concepts. The developed formalization applies mainly a part of the NASA PVS library for abstract algebra specified in the theory algebra.

In this paper, we consider homeomorphisms of compact connected orientable Seifert manifolds with hyperbolic orbifolds, and give some bounds involving the rank and the index of fixed point classes. One consequence is an index bound for fixed point classes. We rely on the classification of 2-orbifolds homeomorphisms and the similar bounds on surfaces.

Let R be a ring with identity, σ be an automorphism, and δ be a derivation of R. In this article, we present some necessary and sufficient conditions on R such that the skew polynomial rings R[x, x −1; σ], R((x; σ)), and R[x; δ] are prime or semiprime.

In this article we prove the following result. Let n ≥ 1 be some fixed integer, and let R be a prime ring with 2n ≤ char(R) ≠ 2. Suppose there exists an additive mapping T: R → R satisfying the relation for all x ∈ R. In this case, T is of the form 4T(x) = qx + xq for all x ∈ R, where q is some fixed element from the symmetric Martindale ring of quotients. This result makes it possible to solve some functional equations in prime rings with involution which are related to bicircular projections.

We find a method to recognize the characteristic of a simple group of Lie type G from its Dirichlet polynomial P G ( s ) . This is enough to complete the proof of the following statement: if G is a simple group, H is a finite group and P G ( s ) = P H ( s ) , then H / Frat ( H ) ≅ G .

It is shown that if the group ring RQ8 of the quaternion group Q8 of order 8 over an integral domain R is duo, then R is a field for the following cases: (1) char R ≠ 0, and (2) char R = 0 and S ⊆ R ⊆ KS, where S is a ring of algebraic integers and KS is its quotient field. Hence, we confirm that the field ℚ of rational numbers is the smallest integral domain R of characteristic zero such that RQ8 is duo. A non-field integral domain R of characteristic zero for which RQ8 is duo is also identified. Moreover, we give a description of when the group ring RG of a torsion group G is duo.

Using Fourier�Mukai transforms, we prove some results about the ring of unipotent vector bundles on elliptic curves in positive characteristics. This ring was determined by Atiyah in characteristic zero, who showed that it is a polynomial ring in one variable. It turns out that the situation in characteristic p > 0 is completely different and rather bizarre: the ring is nonnoetherian and contains a subring whose spectrum contains infinitely many copies of Spec(Z), which are glued with successively higher and higher infinitesimal identification at the point corresponding to the prime p.

With α irrational the graph of [jα] against integer j, displays interesting patterns of self-matching. This is best seen by comparing the Bernoulli (characteristic Sturmian) or difference sequence 〈βj〉, term by term with the Bernoulli sequence displaced by k terms 〈βj−k〉, where βj=[(j+1)α]−[jα].It is shown that the fraction of such self-matching is the surprisingly simple M(k)=max(|1−2{α}|,|1−2{kα}|).Of particular interest is the graph of M(k) against k as it is seen to exhibit an unexpected Moiré pattern obtained simply by folding the lower half of the graph of {kα} over the upper half.

Using the theory of Witt vectors, we define ring structures on several well-known groups of arithmetic functions, which in another guise are formal Dirichlet series. The set of multiplicative arithmetic functions over a commutative ring R is shown to have a unique functorial ring structure for which the operation of addition is Dirichlet convolution and the operation of multiplication restricted to the completely multiplicative functions coincides with point-wise multiplication. The group of additive arithmetic functions over R also has a functorial ring structure. In analogy with the ghost homomorphism of Witt vectors, there is a functorial ring homomorphism from the ring of multiplicative functions to the ring of additive functions that is an isomorphism if R is a Q -algebra. The group of rational arithmetic functions, that is, the group generated by the completely multiplicative functions, forms a subring of the ring of multiplicative functions. The latter ring has the structure of a Bin ( R ) -algebra, where Bin ( R ) is the universal binomial ring equipped with a ring homomorphism to R. We use this algebra structure to study the order of a rational arithmetic function, as well the powers f α for α ∈ Bin ( R ) of a multiplicative arithmetic function f. For example, we prove new results about the powers of a given multiplicative arithmetic function that are rational. Finally, we apply our theory to the study of the zeta function of a scheme of finite type over Z .

International audience; This paper deals with adaptive observers designs for some class of uniformly observable systems. Indeed, an adaptive observer is proposed for the on-line estimation of the state and the unknown parameters involved in the system dynamics as well as in the output expression. It is shown that the equations of the proposed observer can be derived using two approaches, namely the adaptive approach and extended-high gain observer-based approach. The observer design is detailed under both approaches and the equivalence between the underlying observer equations is established.

Let m be any positive integer, R be a ring with identity, be the matrix ring of all m by m matrices eve. R and be the multiplicative group of all n by n nonsingular matrices in . In this pape., the following are investigated: (1) for any pairwise coprime ideals in a ring R, is isomorphic to and is isomorphic to (2) In particular, if R is a finite ring with identity, then the order of can be computed.

Let M n ( F ) be the space of all n×n matrices over a field F of characteristic not 2, and let P n ( F ) be the subset of M n ( F ) consisting of all n×n idempotent matrices. We denote by Φ n ( F ) the set of all maps from M n ( F ) to itself satisfying A−λB∈P n ( F ) if and only if φ(A)−λφ(B)∈P n ( F ) for every A,B∈M n ( F ) and λ∈ F . It was shown that φ∈Φ n ( F ) if and only if there exists an invertible matrix P∈M n ( F ) such that either φ(A)=PAP−1 for every A∈M n ( F ) , or φ(A)=PATP−1 for every A∈M n ( F ) . This improved Dolinar's result by omitting the surjectivity assumption and extending the complex field to any field of characteristic not 2.

Let R be a prime ring and U(R) the group of units of R. We prove that if U(R) generates R and satisfies a group identity,then R is either a domain or a full matrix ring over a finite field.

A new method for extracting junction parameters of the single diode model is presented. A least squares method approach considers the deviation ∆V=f(I) between the experimental current-voltage (I-V) characteristic and a theoretical arbitrary characteristic. A specific case- the ∆V graph reducing to a straight line–is identified and the knowledge of the slope and of the intercept with the ordinate axis leads to the determination of the junction parameters. The method is applied to the characterization of the emitter-base junction of transistors and the results are discussed.

The notions of F -rational and F -regular rings are defined via tight closure, which is a closure operation for ideals in a commutative ring of positive characteristic. The geometric significance of these notions has persisted, and K. E. Smith proved that F -rational rings have rational singularities. We now ask about the converse implication. The answer to this question is yes and no. For a fixed positive characteristic, there is a rational singularity which is not F -rational, so the answer is no. In this paper, however, we aim to show that the answer is yes in the following sense: If a ring of characteristic zero has rational singularity, then its modulo p reduction is F -rational for almost all characteristic p . This result leads us to the correspondence of F -regular rings and log terminal singularities.

Consider a dynamical system given by a planar differential equation, which exhibits an unstable periodic orbit surrounding a stable periodic orbit. It is known that under random perturbations, the distribution of locations where the system's first exit from the interior of the unstable orbit occurs, typically displays the phenomenon of cycling: The distribution of first-exit locations is translated along the unstable periodic orbit proportionally to the logarithm of the noise intensity as the noise intensity goes to zero. We show that for a large class of such systems, the cycling profile is given, up to a model-dependent change of coordinates, by a universal function given by a periodicised Gumbel distribution. Our techniques combine action-functional or large-deviation results with properties of random Poincar\'e maps described by continuous-space discrete-time Markov chains., Comment: 44 pages, 4 figures

We show that a category A with a coherent ring structure determines a ring spectrum EA whose zeroth space is an additive group completion of the classifying space BA. This contains May's result that a bipermutative category determines an E∞ ring spectrum. It also implies that a braided bimonoidal category determines an E2 ring spectrum. If X is an En-space, we also construct a ring category A whose associated ring spectrum EA is the En ring spectrum QX +, the stable homotopy of X−.

International audience; The structure at infinity and the essential structure are two control theory notions which were first defined for linear, then for the so-called affine, systems. And they were shown to be useful tools for the study of the fundamental problem of noninteracting control. They also appeared as related to the solutions of other important control problems such as disturbance decoupling. Their definitions are however entirely in terms of an algorithm, namely the so-called structure algorithm. The present work proposes new definitions with some advantages: they extend the class of systems from linear and affine systems to systems which may be described by algebraic differential equations, they are not tied to specific algorithms, and finally they provide more information on system structure. Let a system be a set of differential equations in variables which are grouped as m inputs, p outputs and n latent variables. To each input component is attached a rational integer, which, for a single input single output system defined by a single differential equation, is the difference between the order in the output and the order in the input of the differential equation defining the system. The m-tuple of these rational integers is the new structure at infinity of the system. Associated to the structure at infinity is also defined a p-tuple of rational integers representing a new notion of essential structure. The old structure at infinity is shown to be recoverable from the new one. Computations of system structure based upon the suggested definitions are quite complex. The present paper focuses on proofs of algorithms which attempt to reduce the complexity of these computations.

It is proved that the algebra of regular functions on quantum matrices admits a division ring of quotients and that this division ring is a division ring of twisted rational functions. A description is given of the field of central elements in the division ring of rational functions on quantum matrices in the one-parameter and multiparameter cases. In the one-parameter case for of a general form the center is a purely transcendental extension of a field of degree (were is the greatest common divisor of and ) if both numbers and are odd. If at least one of the numbers and is even, then the center is scalar. In the multiparameter case the answer depends upon the parameters ,, . Here the generators of the center are described and it is proved that the center is scalar for the case of even and parameters of a general form. Analogous result are obtained for the division ring of rational functions on a quantum Borel subgroup of .

In studies of population variability, particular attention has to be paid to the selection of a representative sample. The aim of this study was to assess the size of the new representative sample on the basis of the variability of chemical content of the initial sample on the example of a whitebark pine population. Statistical analysis included the content of 19 characteristics (terpene hydrocarbons and their derivates) of the initial sample of 10 elements (trees). It was determined that the new sample should contain 20 trees so that the mean value calculated from it represents a basic set with a probability higher than 95 %. Determination of the lower limit of the representative sample size that guarantees a satisfactory reliability of generalization proved to be very important in order to achieve cost efficiency of the research. [Projekat Ministarstva nauke Republike Srbije, br. OI-173011, br. TR-37002 i br. III-43007]

According to general terminology, a ringRis completely primary if its set of zero divisorsJforms an ideal. LetRbe a finite completely primary ring. It is easy to establish thatJis the unique maximal ideal ofRandRhas a coefficient subringS(i.e.R/Jisomorphic toS/pS) which is a Galois ring. In this paper we give the construction of finite completely primary rings in which the product of any two zero divisors is inSand determine their enumeration. We also show that finite rings in which the product of any two zero divisors is a power of a fixed prime p are completely primary rings with eitherJ2=0or their coefficient subring isZ2nwithn=2or3. A special case of these rings is the class of finite rings, studied in [2], in which the product of any two zero divisors is zero.

An associative ring R with identity is semiperfect if and only if every element of R is a sum of a unit and an idempotent, and R contains no infinite set of orthogonal idempotents. A ring which contains no infinite set of orthogonal idempotents is an exchange ring if and only if every element is a sum of a unit and an idempo-tent

It is shown that a ring homomorphism onH(G), the algebra of analytic functions on a regular regionGin the complex plane, is either linear or conjugate linear provided that the ring homomorphism takes the identity function into a nonconstant function.

Let S be a subring of the ring R. We investigate the question of whether S ∩ U ( R ) = U ( S ) holds for the units. In many situations our answer is positive. There is a special emphasis on the case when R is a full matrix ring and S is a structural subring of R defined by a reflexive and transitive relation.

We prove that the F-signature of an affine semigroup ring of positive characteristic is always a rational number, and describe a method for computing this number. We use this method to determine the F-signature of Segre products of polynomial rings, and of Veronese subrings of polynomial rings. Our technique involves expressing the F-signature of an affine semigroup ring as the difference of the Hilbert-Kunz multiplicities of two monomial ideals, and then using Watanabe's result that these Hilbert-Kunz multiplicities are rational numbers.

Let W be a finite dimensional representation of a linearly reductive group G over a field k. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under G of the symmetric algebra of W has a simple ring of differential operators. In this paper, we show that this is true in prime characteristic. Indeed, if R is a graded subring of a polynomial ring over a perfect field of characteristic p>0 and if the inclusionof R into S splits, then D_k(R) is a simple ring. In the last section of the paper, we discuss how one might try to deduce the characteristic zero case from this result. As yet, however, this is a subtle problem and the answer to the question of Levasseur and Stafford remains open in characteristic zero., 30 pages; Latex file; One minor difference between this version and published version: Incorrect justification for one easy statement in proof of Proposition 3.1.6 corrected

It is conjectured that every ring with a finite maximal subring is finite. We prove this conjecture forPI-rings.

In this chapter we shall explore the problem of correlation between a given prime ring R and a subring of constants of a finite-dimensional Lie algebra L of its derivations. If the characteristic of a ring is equal to zero, then constants of finite-dimensional Lie algebras are related, but weakly, with the basic ring. Let, for instance, be a free algebra with no unit. Let us consider a derivations μ, such that Yμ = Y, Xμ = X. Then its derivation has the only constant, i.e., zero, in which case the Lie algebra generated by μ is one-dimensional. A reasonable restriction which can be imposed in the case of a zero characteristic is that the associative algebra generated by the considered derivations in a ring of endomorphisms of an additive group of the ring R is finite-dimensional. In this case, however, theorem 2.7.8 yields that L consists only of the derivations which are inner for the ring of quotients Q( R), and the problems are reduced to those of centralizers of finite-dimensional subalgebras, which have already been considered in 3.3.

Certain methods of determining the coefficients of the output characteristic of a ring laser are proposed. The coefficient K/sub 1/ was determined with not more than a 10/sup -5/ relative error, and the remaining coefficients which are 3 to 4 orders of magnitude less than K/sub 1/ were determined with a relative error on the order of several percent. The results obtained permit finding separate components of the output characteristic of a ring laser with low errors, as is necessary in a complex investigation of a ring laser. (MHR)

We define the 0—1 Integer Programming Problem in a finite field or finite ring with identity as: given an m × n matrix A and an n × 1 vector b with entries in the ring R, find or determine the non-existence of a 0—1 vector x such that Ax = b. We give an easily implemented enumerative algorithm for solving this problem, along with conditions that spurious solutions occur with probability as small as desired. Finally, we show that the problem is NP-complete if R is the ring of integers modulo r for r ≥ 3. This result suggests that it will be difficult to improve on our algorithm.

The first note shows that if R > T R > T are any two Noetherian rings, then there exists a Noetherian ring B B between R R and T T which has a maximal ideal N N such that grade ( N ) ≤ 1 {\text {grade}}(N) \leq 1 and N ∩ R N \cap R is a maximal ideal. The second note shows that if R R is a Noetherian ring, then there exists a free quadratic integral extension ring B B of R R such that Spec ⁡ ( B ) ≅ Spec ⁡ ( R ) \operatorname {Spec}(B) \cong \operatorname {Spec}(R) and such that if I I is any regular ideal in R R and P 1 ∩ ⋯ ∩ P g {P_1} \cap \cdots \cap {P_g} are prime ideals in R R containing I I , then there exists an ideal J J in B B integrally dependent on I B IB such that the prime ideals corresponding to the P i {P_i} are prime divisors of J n {J^n} for all n ≥ 1 n \geq 1 .

A type of ring extension is considered that was introduced by J. Szendrei and generalizes many familiar examples, including the complex extension of the real field. We give a method for constructing a large class of examples of this type of extension, and show that for some rings all possible examples are obtained by this method. An abstract characterization of the extension is also given, among rings defined on the set product of two given rings. This paper is a sequel to [2 ], in which a class of examples was given of a type of ring extension defined in terms of two functions. Here we exhibit some other functions that may be used to construct such extensions, and show that in certain cases (in particular, when the first ring is an integral domain and the second is a commutative ring with identity), the functions must have a prescribed form. We also characterize this type of ring extension, which Szendrei defined directly by the ring operations, in terms of the manner in which the two given rings are embedded in the extension. The reader is referred to [2] for background. Only the basic definition is repeated here. Let A and B be rings. We define the ring A*B to be the direct sum of A and B as additive groups, with multiplication given by (1) (a, b)(c, d) = (ac + {b, d}, ad + bo-c + bd), where ois a homomorphism from A onto a ring of permutable bimultiplications of B, and {I , *} is a biadditive function from B XB into A satisfying the equations (2) bo-1c, d} = 1b, c)d, (3) 1{b, cd} = {bc, di, (4) {b ?oaC} = {boa, c}, (5) aOabC}c = a{b,c}, and {b,co-a} = {b,c}a, for all aCA and all b, c, deB. As noted in [2], when B has an identity, (4) is redundant. The inverse of a homomorphism 4 will be denoted by O', even when 4 is not a monomorphism. Received by the editors September 2, 1969. AMS Subject Classifications. Primary 1680, 1380; Secondary 1315.

A graph G is called (k,d)^*-choosable if, for every list assignment L with |L(v)|=k for all [email protected]?V(G), there is an L-coloring of G such that every vertex has at most d neighbors having the same color as itself. Let G be a graph embeddable in a surface of nonnegative characteristic. In this paper, we prove: (1) If G contains no k-cycle with a chord for all k=4,5,6, then G is (3,1)^*-choosable; (2) If G contains neither 5-cycle with a chord nor 6-cycle with a chord, then G is (4,1)^*-choosable.