Subset Of Ring Research Articles

A commutativity condition on subsets of rings

Let [Formula: see text] and [Formula: see text]. A ring [Formula: see text] is called a [Formula: see text]-ring if for every [Formula: see text] [Formula: see text]-subsets [Formula: see text] of [Formula: see text] there exist [Formula: see text] and [Formula: see text] such that [Formula: see text]. We give noncommutative examples of such rings, and we discuss finiteness and commutativity.

RADICALS DETERMINED BY SUBSETS OF RINGS

Many radicals are determined by a property that subsets of a ring may satisfy; for example, the subsets of a ring singled out could be m-systems leading to the prime radical. Here this process is formalized. In addition, the conditions that a property of the subsets must satisfy to ensure that the corresponding radical fulfills a variety of criteria are described.

Algebraicity of the radical in local rings

In this paper we continue the study of algebraic subsets of non-commutative local rings. (A subset of a ring is said to be algebraic if there exists a monic polynomial with coefficients from the ring vanishing on the subset.) In particular, we prove that the Jacobson radical of a local ring is an algebraic subset if and only if it is a nil ideal of bounded index.

Extensions of (weakly) null-additive, monotone set functions from rings of subsets to generated algebras

This paper shows that the greatest and least monotone extensions of a null-additive (resp. weakly null-additive), monotone set function from a ring of subsets to the algebra generated by the ring are null-additive (resp. weakly null-additive). In addition, the paper characterizes all the (weakly) null-additive, monotone extensions.

On Some Centre–Like Subsets of Rings

For arbitrary rings R, let Fr(R) = {aeR ''{a, b}2' < 3 for all beR}, and let F*(R) = {a e R' for each beR, either [a, b] = 0 or a2 = b2}. We study the properties of these sets for large classes of rings, and we provide sufficient conditions for these sets to coincide with the centre. Several examples are given.

ON THE EXTENSION OF SET-VALUED SET FUNCTIONS

We extend additive set-valued set functions and normal multimeasures defined on a ring of subsets. We also prove a Carathéodory-Hahn-Kluvanek-type theorem for additive set-valued set functions. Finally, we establish results on the extension of transition multimeasures.

On functional identities and d-free subsets of rings. II

We show that a map in several variables on a prime ring satisfying an identity of polynomial type must be a quasi-polynomial (i.e., a polynomial in noncommutative variables whose coefficients are Martindale centroid valued functions)provided that the ring does not satisfy a standard identity of low degree. Obtained results have applications to the study of Lie maps of prime rings (Lie ideals of prime rings and skew elements of prime rings with involution)and to the study of Lie-admissible algebras and Lie homomorphisms of Lie algebras of Poisson algebras.

The Ubiquitin-conjugating Enzymes UbcH7 and UbcH8 Interact with RING Finger/IBR Motif-containing Domains of HHARI and H7-AP1

Ubiquitinylation of proteins appears to be mediated by the specific interplay between ubiquitin-conjugating enzymes (E2s) and ubiquitin-protein ligases (E3s). However, cognate E3s and/or substrate proteins have been identified for only a few E2s. To identify proteins that can interact with the human E2 UbcH7, a yeast two-hybrid screen was performed. Two proteins were identified and termed human homologue of Drosophila ariadne (HHARI) and UbcH7-associated protein (H7-AP1). Both proteins, which are widely expressed, are characterized by the presence of RING finger and in between RING fingers (IBR) domains. No other overt structural similarity was observed between the two proteins. In vitro binding studies revealed that an N-terminal RING finger motif (HHARI) and the IBR domain (HHARI and H7-AP1) are involved in the interaction of these proteins with UbcH7. Furthermore, binding of these two proteins to UbcH7 is specific insofar that both HHARI and H7-AP1 can bind to the closely related E2, UbcH8, but not to the unrelated E2s UbcH5 and UbcH1. Although it is not clear at present whether HHARI and H7-AP1 serve, for instance, as substrates for UbcH7 or represent proteins with E3 activity, our data suggests that a subset of RING finger/IBR proteins are functionally linked to the ubiquitin/proteasome pathway.

On Additive Maps and Commutativity in Rings

We study additive maps which are skew-commuting or skew-centralizing on appropriate subsets of a ring R; and we investigate commutativity in prime and semiprime rings admitting a nonzero derivation d such that [d(x),d(y)] = 0 for all x,y in some nonzero one-sided ideal. This paper has two main parts. The first, motivated by a recent result of Brešar [3] on triviality of skew-commuting additive maps on prime rings, is a study of additive maps which are skew-commuting or skew-centralizing on subsets of certain rings. The second continues a study, begun years ago by Herstein [7], of prime and semiprime rings R admitting a nonzero derivation d such that d(x)d(y) − d(y)d(x) = 0 for all x, y in a suitably chosen subset of R.

Testing Lattice Conditional Independence Models

The lattice conditional independence (LCI) model N( K ) is defined to be the set of all normal distributions N(0, Σ) on R I such that for every pair L, M ∈ K , x L and x M are conditionally independent given x L ∩ M . Here K is a ring of subsets (hence a distributive lattice) of the finite index set I such that ∅ I ∈ K , while for K ∈ K , x K is the coordinate projection of x ∈ R I onto R K . These LCI models have especially tractable statistical properties and arise naturally in the analysis of non-monotone multivariate missing data patterns and non-nested dependent linear regression models ≡ seemingly unrelated regressions. The present paper treats the problem of testing one LCI model against another, i.e., testing N( K ) vs N( M ) when M is a subring of K . The likelihood ratio test statistic is derived, together with its central distribution, and several examples are presented.

Unbiasedness of the Likelihood Ratio Test for Lattice Conditional Independence Models

The lattice conditional independence (LCI) model N( K ) is defined to be the set of all normal distributions N(0, Σ) on R I such that for every pair L, M ∈ K , x L and x M are conditionally independent given x L ∩ M . Here K is a ring of subsets (hence a distributive lattice) of the finite index set I such that ∅ I ∈ K , while for K ∈ K , x K is the coordinate projection of x ∈ R I onto R K . Andersson and Perlman in the preceding paper derived the likelihood ratio (LR) statistic λ for testing one LCI model against another, i.e., for testing N( K ) vs N( M ) based on a random sample from N(0, Σ), where M is a subring of K . In the present paper the strict unbiasedness of the LR test is established, and related results regarding the distribution of the maximum likelihood estimator of Σ under the LCI model N( K ) are presented.

Branching inset entropies on open domains

The forms of all semisymmetric, branching, multidimensional measures of inset information on open domains are determined. This is done for both the complete and the possibly incomplete partition cases. The key to these results is to find the general solution of the functional equation $$(*)\Delta \left( {\begin{array}{*{20}c} {E,F} \\ {p,q} \\ \end{array} } \right) + \Delta \left( {\begin{array}{*{20}c} {E \cup F,G} \\ {p + q,r} \\ \end{array} } \right) = \Delta \left( {\begin{array}{*{20}c} {E,G} \\ {p,r} \\ \end{array} } \right) + \Delta \left( {\begin{array}{*{20}c} {E \cup G,F} \\ {p + r,q} \\ \end{array} } \right)$$ for all(p, q, r) ∈ D3:={(p1, p2, p3)∣pi∈]0, 1[k(i = 1,2,3),p1+ p2+ p3∈]0, 1[k} and all (E, F, G) for which there exists anH with (E, F, G, H) ∈ ℒ4:={(E1,⋯,E4)∣Ei ∈ ℛ\{O},E1∩Ej = O for alli ≠ j}. Here, ℛ is a ring of subsets of a given universal set. Functional equation (*) is solved by use of the general solution of the generalized characteristic equation of branching information measures.

On some center-like subsets of rings

On some center-like subsets of rings

Families of cuts with the MFMC-property

A family ℱ of cuts of an undirected graphG=(V, E) is known to have the weak MFMC-property if (i) ℱ is the set ofT-cuts for someT⊆V with |T| even, or (ii) ℱ is the set of two-commodity cuts ofG, i.e. cuts separating any two distinguished pairs of vertices ofG, or (iii) ℱ is the set of cuts induced (in a sense) by a ring of subsets of a setT⊆V. In the present work we consider a large class of families of cuts of complete graphs and prove that a family from this class has the MFMC-property if and only if it is one of (i), (ii), (iii).

Characterization of a Class of Equicontinuous Sets of Finitely Additive Measures with an Application to Vector Valued Borel Measures

Let V denote a ring of subsets of an abstract space X, let R + denote the nonnegative reals, and let N denote the set of positive integers. We denote by C(V) the space of all subadditive and increasing functions, from the ring V into R +, which are zero at the empty set. The space C(V) is called the space of contents on the ring V and elements are referred to as contents. A sequence of sets An ∊ V, n ∊ N is said to be dominated if there exists a set B ∊ V such that An ⊆ B, for n = 1, 2, A content p ∊ C(V) is said to be Rickart on the ring V if lim n p(An ) = 0 for each dominated, disjoint sequence An ∊ V, n ∊ N.

A Note on Equicontinuous Families of Volumes With an Application to Vector Measures

Let V denote a ring of subsets of an abstract space X, let R denote the real numbers, and let N denote the positive integers. Denote by a(V, R) (respectively ca(V, R)) the space of real valued, finitely additive (respectively countably additive) functions on the ring V and denote by ab(V, R) the subspace consisting of those members of the space a(V, R) with finite variation on each set in the ring V. Members of the space a(V, R) are referred to as charges and members of the space ab(V, R) are referred to as locally bounded charges. We denote by cab(V, R) the intersection of the spaces ab(V, R) and ca(V, R).

On the decomposition of measures with applications to continuum physics

In this note we show that some concepts of modern measure theory may be relevant to the formulation of theories in continuum physics. In particular we observe that a large class of measure spaces possesses a decomposition property which is useful in applications. This decomposition property is obtained as a corollary to a theorem of Kelley [10] concerning the "Lebesgue decomposition property," which result is based upon the investigations of Segal [3]. The mathematical notation and definitions (following Kelley [10]), are presented first; then a Lemma and the Decomposition are given. The relationship between these results and physical theories is presented in a series of remarks at the end . As a preliminary we establish some useful definitions and notation. I Let be a set and let sg/a be a ring of subsets of M whose union is ~ . We further assume that d / ~ has the property: if {d~ } is a sequence of sets in d¢ "~ such that if ~¢. _ d E d¢ '~, for all n and some d in d /~ , then U . d . ~ d¢ 'a~. By a measure v in we mean a non-negative, finite valued, countably additive set function defined on ~ ' a with the following additional property: if {~ ' . } is a disjoint sequence in ~.¢/a such that g . v ( d . ) converges, then U . d . a jfa~.2 A subfamily q / i n ~ ' a is called a a-ideal in d¢ 'a~ if and only if it is a ring of sets, the intersection of a member of q /wi th a member o f ~ / a is a member of q/, and any member of~¢¢ '~ which is a countable union of sets in ~ is itself a set in q/. A set ~ _ ~ is (locally) measurable if and only if ~¢ n ~ e J / fa for all ~¢ in J/¢'~. We denote the class of measurable sets in ~ by ~//¢ and observe that J g is a or-algebra of subsets of ~ containing ~ ,a .3 I f~¢ and cg are in J r , we write ~¢Ro~ if and only i f c ~ ~ ¢ contains no set