Cohen's Theorem Research Articles

Geometry of Banach algebra [formula omitted] and the bidual of [formula omitted

This article is intended towards the study of the bidual of generalized group algebra L1(G,A) equipped with two Arens products, where G is any locally compact group and A is a Banach algebra. We show that the left topological center of (L1(G)⊗ˆA)⁎⁎ is a Banach L1(G)-module if G is abelian. Further it also holds permanence property with respect to the unitization of A. We then use this fact to extend the remarkable result of A.M. Lau and V. Losert [7], about the topological center of L1(G)⁎⁎ being just L1(G), to the reflexive Banach algebra valued case using the theory of vector measures. We further explore pseudo-center of L1(G,A)⁎⁎ for non-reflexive Banach algebras A and give a partial characterization for elements of pseudo-center using the Cohen's factorization theorem. In the running we also observe few consequences when A holds the Radon-Nikodym property and weak sequential completeness.

Cohen's theorem for a class of Noetherian semirings

Cohen's theorem for a class of Noetherian semirings

On $ S $-principal right ideal rings

<abstract><p>Let $ S $ be a multiplicative subset of a ring $ R $. A right ideal $ A $ of $ R $ is referred to as <italic>$ S $-principal</italic> if there exist an element $ s \in S $ and a principal right ideal $ aR $ of $ R $ such that $ As \subseteq aR \subseteq A $. A ring is referred to as an $ S $-<italic>principal right ideal ring</italic> ($ S $-PRIR) if every right ideal is $ S $-principal. This paper examines $ S $-PRIRs, which extend the notion of principal right ideal rings. Various examples, including several extensions of $ S $-PRIRs are investigated, and some practical results are proven. A noncommutative $ S $-PRIR that is not a principal right ideal ring is found, and the $ S $-variants of the Eakin-Nagata-Eisenbud theorem and Cohen's theorem for $ S $-PRIRs are proven.</p></abstract>

A fluctuation theorem for Floquet quantum master equations

We present a fluctuation theorem for Floquet quantum master equations. This is a detailed version of the famous Gallavotti–Cohen theorem. In contrast to the latter theorem, which involves the probability distribution of the total heat current, the former involves the joint probability distribution of positive and negative heat currents and can be used to derive the latter. A quantum two-level system driven by a periodic external field is used to verify this result.

Testing the Steady-State Fluctuation Relation in the Solar Photospheric Convection.

The turbulent thermal convection on the Sun is an example of an irreversible non-equilibrium phenomenon in a quasi-steady state characterized by a continuous entropy production rate. Here, the statistical features of a proxy of the local entropy production rate, in solar quiet regions at different timescales, are investigated and compared with the symmetry conjecture of the steady-state fluctuation theorem by Gallavotti and Cohen. Our results show that solar turbulent convection satisfies the symmetries predicted by the fluctuation relation of the Gallavotti and Cohen theorem at a local level.

Time-reversal symmetric Crooks and Gallavotti–Cohen fluctuation relations in driven classical Markovian systems

In this paper, we address an important question of the relationship between fluctuation theorems for the dissipated work with general finite-time (like Jarzynski equality and Crooks relation) and infinite-time (like Gallavotti–Cohen theorem) drive protocols and their time-reversal symmetric versions. The relations between these kinds of fluctuation relations are uncovered based on the examples of a classical Markovian N-level system. Further consequences of these relations are discussed with respect to the possible experimental verifications.

On the difference between permutation polynomials

The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if p>(d2−3d+4)2, then there is no complete mapping polynomial f in Fp[x] of degree d≥2. For arbitrary finite fields Fq, a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of f.Cohen, Mullen and Shiue generalized the Chowla–Zassenhaus–Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if f and f+g are both permutation polynomials of degree d≥2 over Fp, with p>(d2−3d+4)2, then the degree k of g satisfies k≥3d/5, unless g is constant. In this article, assuming f and f+g are permutation polynomials in Fq[x], we give lower bounds for the Carlitz rank of f in terms of q and k. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials f of Carlitz rank n≥1 that if f+xk is a permutation over Fq, with gcd⁡(k+1,q−1)=1, then k≥(q−n)/(n+3).

The multiplicative ideal theory of Leavitt path algebras

It is shown that every Leavitt path algebra L of an arbitrary directed graph E over a field K is an arithmetical ring, that is, the distributive law A∩(B+C)=(A∩B)+(A∩C) holds for any three two-sided ideals of L. It is also shown that L is a multiplication ring, that is, given any two ideals A,B in L with A⊆B, there is always an ideal C such that A=BC, an indication of a possible rich multiplicative ideal theory for L. Existence and uniqueness of factorization of the ideals of L as products of special types of ideals such as prime, irreducible or primary ideals is investigated. The irreducible ideals of L turn out to be precisely the primary ideals of L. It is shown that an ideal I of L is a product of finitely many prime ideals if and only the graded part gr(I) of I is a product of prime ideals and that I/gr(I) is finitely generated with a generating set of cardinality no more than the number of distinct prime ideals in the prime factorization of gr(I). As an application, it is shown that if E is a finite graph, then every ideal of L is a product of prime ideals. The same conclusion holds if L is two-sided artinian or two-sided noetherian. Examples are constructed verifying whether some of the well-known theorems in the ideal theory of commutative rings such as the Cohen's theorem on prime ideals and the characterizing theorem on ZPI rings hold for Leavitt path algebras.

Existence Of Coefficient Subring for Transcendental Extension Ring

As a consequence of Cohen's structure Theorem for complete local rings that every _nite commutative ring R of characteristic pn contains a unique special primary subring R0 satisfying R/J(R) = R0/pR0: Cohen called R0 the coe_cient subring of R. In this paper we will study the case when the ring is a transcendental extension local artinian duo ring R; we proved that even in this case R will has a commutative coe_cient subring.

On Lie Nilpotent Rings and Cohen's Theorem

We study certain (two-sided) nil ideals and nilpotent ideals in a Lie nilpotent ring R. Our results lead us to showing that the prime radical rad(R) of R comprises the nilpotent elements of R, and that if L is a left ideal of R, then L + rad(R) is a two-sided ideal of R. This in turn leads to a Lie nilpotent version of Cohen's theorem, namely if R is a Lie nilpotent ring and every prime (two-sided) ideal of R is finitely generated as a left ideal, then every left ideal of R containing the prime radical of R is finitely generated (as a left ideal). For an arbitrary ring R with identity we also consider its so-called n-th Lie center Z n (R), n ≥ 1, which is a Lie nilpotent ring of index n. We prove that if C is a commutative submonoid of the multiplicative monoid of R, then the subring ⟨Z n (R) ∪ C⟩ of R generated by the subset Z n (R) ∪ C of R is also Lie nilpotent of index n.

$r$-ideals in commutative rings

In this article we introduce the concept of $r$-ideals in commutative rings (note: an ideal $I$ of a ring $R$ is called $r$-ideal, if $ab\in I$ and ${\rm Ann}(a)=(0)$ imply that $b\in I$ for each $a,b\in R$). We study and investigate the behavior of $r$-ideals and compare them with other classical ideals, such as prime and maximal ideals. We also show that some known ideals such as $z^\circ$-ideals are $r$-ideals. It is observed that if $I$ is an $r$-ideal, then so too is a minimal prime ideal of $I$. We naturally extend the celebrated results such as Cohen's theorem for prime ideals and the Prime Avoidance Lemma to $r$-ideals. Consequently, we obtain interesting new facts related to the Prime Avoidance Lemma. It is also shown that $R$ satisfies property $A$ (note: a ring $R$ satisfies property $A$ if each finitely generated ideal consisting entirely of zerodivisors has a nonzero annihilator) if and only if for every $r$-ideal $I$ of $R$, $I[x]$ is an $r$-ideal in $R[x]$. Using this concept in the context of $C(X)$, we show that every $r$-ideal is a $z^\circ$-ideal if and only if $X$ is a $\partial$-space (a space in which the boundary of any zeroset is contained in a zeroset with empty interior). Finally, we observe that, although the socle of $C(X)$ is never a prime ideal in $C(X)$, the socle of any reduced ring is always an $r$-ideal.

COMPUTATIONAL ALGORITHM FOR AN MP3 (MOORE-PENROSE 3) MATRIX RING AND FORMULATION OF MP3 EQUIVALENCE

The structure theorems for complete local rings are largely due to Irving S. Cohen. The beginning part of Cohen's structure theorem for the equicharacteristic case is A complete local ring which has the same characteristic as its residue field contains a coefficient field, see (1) The second part of Cohen's structure theorem conveys essentially that in the equicharacteristic case, a complete local ring is a homormorphic image of a formal power series ring over the residue field. This paper proposes the algebraic structure of an n × n matrix ring Mn(R) (n ≥ 1) where R is a formal power series in one variable over a field modulo a power of its maximal ideal. The structure of this special class of matrix rings is derived using the notion of a pseudoinverse and a generalization of one of the Moore-Penrose conditions. Afterwards, some theoretical properties are given to establish the equivalence of MP3 (Moore-Penrose 3) rings that are becoming more exciting in theoretical algebraic investigations for their symmetry inducing capability.

Coefficients of convergent multiple Walsh-Paley series

The paper is concerned with the behaviour of the coefficients of multiple Walsh-Paley series that are cube convergent to a finite sum. It is shown that even an everywhere convergent series of this kind may contain coefficients with numbers from a sufficiently large set that grow faster than any preassigned sequence. By Cohen's theorem, this sort of thing cannot happen for multiple trigonometric series that are cube convergent on a set of full measure — their coefficients cannot grow even exponentially. Null subsequences of coefficients are determined for multiple Walsh-Paley series that are cube convergent on a set of definite measure.Bibliography: 18 titles.

Gallavotti–Cohen Theorem, Chaotic Hypothesis and the Zero-Noise Limit

The Fluctuation Relation for a stationary state, kept at constant energy by a deterministic thermostat—the Gallavotti–Cohen Theorem— relies on the ergodic properties of the system considered. We show that when perturbed by an energy-conserving random noise, the relation follows trivially for any system at finite noise amplitude. The time needed to achieve stationarity may stay finite as the noise tends to zero, or it may diverge. In the former case the Gallavotti–Cohen result is recovered, while in the latter case, the crossover time may be computed from the action of ‘instanton’ orbits that bridge attractors and repellors. We suggest that the ‘Chaotic Hypothesis’ of Gallavotti, Cohen and Ruelle can thus be reformulated as a matter of stochastic stability of the measure in trajectory space. In this form this hypothesis may be directly tested.

A generalization of the classical Krull dimension for modules

In this article, we introduce and study a generalization of the classical Krull dimension for a module M R . This is defined to be the length of the longest strong chain of prime submodules of M (defined later) and, denoted by Cl.K.dim ( M ) . This notion is analogous to that of the usual classical Krull dimension of a ring. This dimension, Cl.K.dim ( M ) exists if and only if M has virtual acc on prime submodules; see Section 2. If R is a ring for which Cl.K.dim ( R ) exists, then for any left R-module M, Cl.K.dim ( M ) exists and is no larger than Cl.K.dim ( R ) . Over any ring, all homogeneous semisimple modules and over a PI-ring (or an FBN-ring), all semisimple modules as well as, all Artinian modules with a prime submodule lie in the class of modules with classical Krull dimension zero. For a multiplication module over a commutative ring, the notion of classical Krull dimension and the usual prime dimension coincide. This yields that for a multiplication module M, Cl.K.dim ( M ) exists if and only if M has acc on prime submodules. As an application, we obtain a nice generalization of Cohen's Theorem for multiplication modules. Also, PI-rings whose nonzero modules have zero classical Krull dimension are characterized.

A Simple Proof of Cohen's Theorem

(2005). A Simple Proof of Cohen's Theorem. The American Mathematical Monthly: Vol. 112, No. 9, pp. 825-826.

A Simple Proof of Cohen's Theorem

A Simple Proof of Cohen's Theorem

Operads and quantum gravity

In this article we try to explain and extend a statement due to Maxim Kontsevich back in 1999 that the Holography Principle in physics should be related to the (higher dimensional) Deligne Conjecture in mathematics. This seems to suggest that the little d -discs operad (or equivalently the notion of a d -algebra) gives a new way to understand the mathematical aspects of quantum gravity using holography. The strategy is as follows: we would like to learn something about quantum gravity in ( d + 1) dimensions: we use holography to reduce our original problem to a CFT in d -dimensions. The deep origin of this dimensional reduction lies on the fact that it is the area and not the volume which appears in the formula giving the entropy of black holes as described long ago by Hawking. Then we use d -algebras (i.e. the little d -discs operad) to study our d -dim CFT. The possible relation between d -dim CFT and d -algebras comes from the lesson we have learnt from strings (namely the 2-dim CFT case): the space of physical states in closed string field theory (i.e. the BRST cohomology) has a natural Gerstenhaber algebra structure and this by Cohen's theorem is related to the little 2-discs operad. The proposal then is that the relation might hold in higher than 2 dimensions. This approach is algebraic although it would have been much more satisfactory if we could generalise Segal's geometric approach to CFT in higher than 2 dimensions. Hopefully the article is mathematically self-contained.

Hilbert's basis theorem for [formula omitted]-radical ideals

We show that under some assumptions the analogues of Hilbert's basis theorem and Cohen's theorem hold for the ∗-ideals in a commutative ring with radical operation ∗.

Electric fields on a surface of constant negative curvature

A one-parameter family of time reversible Anosov flows is studied; physically, it describes a particle moving on a surface of constant negative curvature under the action of an electric field (corresponding to an automorphic form) and of a ‘thermostatting’ force (given by Gauss's least-constraint principle). We show that the flows are dissipative, in the sense that the average volume contraction rate is positive and the Sinai–Ruelle–Bowen measure is singular with respect to the volume: therefore they verify the assumptions for the validity of the continuous time version of Gallavotti–Cohen's fluctuation theorem for the large fluctuations of the average volume contraction rate. If several independent electric fields are considered, it makes sense to ask for the validity of Onsager's reciprocity: we show, by explicitly computing the relevant transport coefficents, that it is indeed obeyed.