Lie *-Nilpotence of Group Rings

Let KG be the group ring of a group G over a field K. Let * be an involution of a group G extended linearly to the group ring KG. Suppose that G is a torsion group without 2-elements and K is a field with characteristic different from 2. We prove that KG is Lie *-nilpotent if and only if KG is Lie nilpotent.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

On FC-Purity and I-Purity of Modules and Köthe Rings

In this article, several characterizations of certain classes of rings via FC-purity and I-purity are considered. Among others results, it is shown that every I-pure injective left R-module is projective if and only if every FC-pure projective left R-module is injective, if and only if, R is a semisimple ring. In particular, the structures of FC-pure projective and I-pure projective modules over a left Artinian ring are completely described. Also, it is shown that every left R-module is FC-pure projective if and only if every indecomposable left R-module is a finitely presented cyclic R-module, if and only if, R is a left Köthe ring. Finally, we introduce FC-pure flatness and I-pure flatness of modules and several characterizations of these notions are given. In particular, we show that a commutative ring R is quasi-Frobenius if and only if R is an Artinian ring and I-pure injective, if and only if, R is an Artinian ring and the injective envelope E(R) is an FC-pure projective R-module.     

On the Weak Order of Orthogonal Groups

A complete lattice structure is defined on the underlying set of the orthogonal group of a real Euclidean space, by a construction analogous to that of the weak order of Coxeter systems in terms of root systems. This produces a complete rootoid in the sense of Dyer, with the orthogonal group as underlying group. It is shown that this complete lattice has a saturation property which is used along with other properties of the lattice to characterize the maximal totally ordered subsets of the lattice as collections of initial sections with respect to a total ordering on the positive roots.     

Right Centralizers of Semiprime Rings

Let R be a semiprime ring with Q _ml (R) the maximal left ring of quotients of R. Suppose that T: R → Q _ml (R) is an additive map satisfying T(x ²) = xT(x) for all x ∈ R. Then T is a right centralizer; that is, there exists a ∈ Q _ml (R) such that T(x) = xa for all x ∈ R.     

On the Finiteness of Noetherian Rings with Finitely Many Regular Elements

Let R be a left Noetherian ring and ZD(R) be the set of all zero-divisors of R. In this paper, it is shown that if R \ ZD(R) is finite, then R is finite.     

Smarandache Vertices of the Graphs Associated to the Commutative Rings

Let R be a commutative ring with identity 1 ≠ 0. A nonzero element a in R is said to be a Smarandache zero-divisor if there exist three different nonzero elements x, y, and b (≠ a) in R such that ax = ab = by = 0, but xy ≠ 0. We will generalize this notion to the Smarandache vertex of an arbitrary simple graph and characterize the Smarandache zero-divisors of commutative rings (resp. with respect to an ideal) via their associated zero-divisor graphs. We illustrate them with examples and prove some interesting results about them.     

Ideal Structure of Leavitt Path Algebras with Coefficients in a Unital Commutative Ring

For a (countable) graph E and a unital commutative ring R, we analyze the ideal structure of the Leavitt path algebra L _R (E) introduced by Mark Tomforde. We first modify the definition of basic ideals and then develop the ideal characterization of Mark Tomforde. We also give necessary and sufficient conditions for the primeness and the primitivity of L _R (E), and we then determine prime graded basic ideals and left (or right) primitive graded ideals of L _R (E). In particular, when E satisfies Condition (K) and R is a field, they imply that the set of prime ideals and the set of primitive ideals of L _R (E) coincide.     

Robustifying Generalized Linear Mixed Models Using a New Class of Mixtures of Multivariate Polya Trees

In applied sciences, generalized linear mixed models have become one of the preferred tools to analyze a variety of longitudinal and clustered data. Due to software limitations, the analyses are often restricted to the setting in which the random effects terms follow a multivariate normal distribution. However, this assumption may be unrealistic, obscuring important features of among-unit variation. This work describes a widely applicable semiparametric Bayesian approach that relaxes the normality assumption by using a novel mixture of multivariate Polya trees prior to define a flexible nonparametric model for the random effects distribution. The nonparametric prior is centered on the commonly used parametric normal family. We allow this parametric family to hold only approximately, thereby providing a robust alternative for modeling. We discuss and implement practical procedures for addressing the computational challenges that arise under this approach. We illustrate the methodology by applying it to real-life examples.

Supplemental materials for this paper are available online.     

A Penalized Likelihood Approach to Rotation of Principal Components

A new paradigm for enhancing the interpretability of principal components through rotation is presented within the framework of penalized likelihood. The rotated components are computed as the maximizers of a Gaussian-based profile log-likelihood function plus a penalty term defined by a standard rotation criterion. This method enjoys a number of advantages over other methods for principal component rotation, notably (1) the rotation specifically targets ill-defined principal components, which may benefit the most from rotation, and (2) the connection with likelihood allows assessment of the fidelity of the rotated components to the data, thereby guiding the choice of penalty parameter. The method is illustrated with an application to a small functional dataset. Efficient computation of the penalized likelihood solution is possible using recently developed algorithms for optimization under orthogonality constraints.