On the Automorphism Group of the First Weyl Algebra

Let A ₁: = 𝕜[t, ?] be the first algebra over a field 𝕜 of characteristic zero. Let Aut_𝕜(A ₁) be the automorphism group of the ring A ₁. One can associate to each right ideal I of A ₁ a subgroup of Aut_𝕜(A ₁) called the isomorphism subgroup of I. In this article, we show that each such isomorphism subgroup is equal to its normalizer. For that, we study when the isomorphism subgroup of a right ideal of A ₁ contains a given isomorphism subgroup.     

Derivations of a Restricted Weyl-Type Algebra Containing the Polynomial Ring

Buchsbaum and Eisenbud [3 Buchsbaum , D. , Eisenbud , D. ( 1977 ). Algebra structures for finite free resolutions and some structure theorems for ideals of codimension 3 . Amer. J. Math. 99 : 447 – 485 .[Crossref], [Web of Science ®] , [Google Scholar]] proved a structure theorem for Gorenstein ideals of grade 3 which claims that every Gorenstein ideal of grade 3 is generated by the submaximal order pfaffians of an alternating matrix. In this article, we characterize some classes of grade 3 perfect ideals which are linked to an almost complete intersection of even type. Furthermore, these structure theorems give us relations between the Gorenstein sequences and the Hilbert functions of the grade 3 perfect ideals that we characterized.     

Primes and Right Ideals in Right Cones

A right cone H in a group G is a submonoid of G that generates G and aH ? bH for a, b ? H implies bH ? aH. With any right ideal I ≠ H of H a completely prime ideal P _r (I) of H is associated and the set 𝒫(I) of right ideals I′ of H with the same associated prime ideal P′ =P _r (I) is determined if P′·? P″ is a right invariant segment in H. The set 𝒫(I) is also described if P _r (I) is a limit prime.     

Reflexive Property of Rings

Mason introduced the reflexive property for ideals, and then this concept was generalized by Kim and Baik, defining idempotent reflexive right ideals and rings. In this article, we characterize aspects of the reflexive and one-sided idempotent reflexive properties, showing that the concept of idempotent reflexive ring is not left-right symmetric. It is proved that a (right idempotent) reflexive ring which is not semiprime (resp., reflexive), can always be constructed from any semiprime (resp., reflexive) ring. It is also proved that the reflexive condition is Morita invariant and that the right quotient ring of a reflexive ring is reflexive. It is shown that both the polynomial ring and the power series ring over a reflexive ring are idempotent reflexive. We obtain additionally that the semiprimeness, reflexive property and one-sided idempotent reflexive property of a ring coincide for right principally quasi-Baer rings.     

Right Nuclei of Quasigroup Extensions

The aim of this article is the study of right nuclei of quasigroups with right unit element. We investigate an extension process in this category of quasigroups, which is defined by a slight modification of non-associative Schreier-type extensions of groups or loops. The main results of the article give characterizations of quasigroup extensions satisfying particular nuclear conditions. We apply these results for constructions of right nuclear quasigroup extensions with right inverse property.     

On Right Alternative Superalgebras

Simple right alternative superalgebras which have a simple algebra as even part and, as odd part, an irreducible bimodule over the even part are investigated. Under these conditions, superalgebras with one dimensional even part are classified, as well as superalgebras having M ₂(F) as even part and a unital irreducible bimodule over M ₂(F) of dimension less than or equal to 6 as odd part. It is shown that there is only a unique non alternative simple right alternative superalgebra of the first type and, for the second type, there is a infinite family depending on a single parameter.     

On subgroups of residually nilpotent metabelian groups

ABSTRACT

We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006 Faith , C. ( 2006 ). Factor rings of pseudo-Frobenius rings . J. Algebra and Its Applications 6 :(to appear). [CSA] [Web of Science ®] , [Google Scholar]). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5).

These results also generalize results of the author in Faith (1985 Faith , C. ( 1985 ). The maximal regular ideal of self-injective and continuous rings splits off . Arch. Math. 44 : 511 – 521 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring.

The results are applied in Section 6 to extend theorems of Faith (1996 Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) characterizing VNR rings, and, as the title of Faith (1996 Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) suggests, extend the conjecture of Shamsuddin.     

Minimizing the area of a Pareto confidence region

A constrained optimization problem is formulated and solved in order to determine the smallest confidence region for the parameters of the Pareto distribution in a proposed family of sets. The objective function is the area of the region, whereas the constraints are related to the required confidence level. Explicit expressions for the area and confidence level of a given region are first deduced. An efficient procedure based on minimizing the corresponding Lagrangian function is then presented to solve the nonlinear programming problem. The process is valid when some of the smallest and largest observations have been discarded or censored, i.e., both single (right or left) and double censoring are allowed. The optimal Pareto confidence region is derived by simultaneously solving three (four) nonlinear equations in the right (double) censoring case. In most practical situations, Newton’s method with the balanced set as the starting point only needs a few iterations to find the global solution. In general, the reduction in area of the optimal Pareto region with respect to the balanced set is considerable if the sample size, n, is small or moderately large, which is usual in practice. This reduction is sometimes impressive when n is quite small and the censoring degree is fairly high. Two numerical examples regarding component lifetimes and fire claims are included for illustrative and comparative purposes.     

Eventual Right Disfocality on Time Scales

Eventual right disfocality of a dynamic equation y j j ( t )+ p 1 ( t ) y j ( t )+ p 2 ( t ) y ( t )=0, t ] T is established where T is a time scale. The results are obtained by imposing integrability conditions on p 1 and p 2 .