On Monoid Rings Over Nil Armendariz RingAlhevaz, A.; Moussavi, A.
doi: 10.1080/00927872.2012.657382pmid: N/A
Given a ring R and a monoid M, we study the concept of so called nil-Armendariz ring relative to a monoid, which is a common generalization of nil-Armendariz rings and Armendariz rings relative to a monoid. It is done by considering the nil-Armendariz condition on a monoid ring R[M] instead of the polynomial ring R[x]. We prove that several properties transfer between R and the monoid ring R[M], in case R is nil M-Armendariz ring. We resolve the structure of nil M-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be nil M-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the special cases. In particular, we prove that every NI-ring is nil M-Armendariz, for any unique product monoid M. We also classify which of the standard nilpotence properties on polynomial rings pass to monoid rings. We provide various examples and classify how the nil M-Armendariz rings behaves under various ring extensions.
The Birman–Murakami–Wenzl Algebras of Type D nCohen, Arjeh M.; Gijsbers, Dié A. H.; Wales, David B.
doi: 10.1080/00927872.2012.678955pmid: N/A
The Birman–Murakami–Wenzl algebra (BMW algebra) of type D n is shown to be semisimple and free of rank (2 n + 1)n!! − (2 n−1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n − 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ℤ[δ±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D n is a subalgebra of the BMW algebra of the same type.
The Proper Class Generated by Weak SupplementsAlizade, Rafail; Demirci, Yılmaz M.; Durğun, Yılmaz; Pusat, Dilek
doi: 10.1080/00927872.2012.699567pmid: N/A
We show that, for hereditary rings, the smallest proper classes containing respectively the classes of short exact sequences determined by small submodules, submodules that have supplements and weak supplement submodules coincide. Moreover, we show that this class can be obtained as a natural extension of the class determined by small submodules. We also study injective, projective, coinjective and coprojective objects of this class. We prove that it is coinjectively generated and its global dimension is at most 1. Finally, we describe this class for Dedekind domains in terms of supplement submodules.
Verbal Subgroups of T r (∞, K)Słowik, R.
doi: 10.1080/00927872.2012.701681pmid: N/A
In this note we investigate the verbal subgroups of T r (∞, K), the group of upper triangular, infinite, row-finite matrices over a field K containing more than two elements. The main result of the work states that every fully invariant subgroup of T r (∞, K), which is contained in its unitriangular subgroup, coincides with some term of the lower central series of T r (∞, K). Using this fact, we describe the verbal subgroups of this group. Moreover, we prove some facts concerning the verbal width of T r (∞, K). In particular, we state that this width is finite.
On the Automorphism Group of the First Weyl AlgebraKouakou, M. K.; Tchoudjem, A.
doi: 10.1080/00927872.2012.704463pmid: N/A
Let A 1: = 𝕜[t, ∂] be the first algebra over a field 𝕜 of characteristic zero. Let Aut𝕜(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 Aut𝕜(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.
On the Annihilator Graph of a Commutative RingBadawi, Ayman
doi: 10.1080/00927872.2012.707262pmid: N/A
Let R be a commutative ring with nonzero identity, Z(R) be its set of zero-divisors, and if a ∈ Z(R), then let ann R (a) = {d ∈ R | da = 0}. The annihilator graph of R is the (undirected) graph AG(R) with vertices Z(R)* = Z(R)∖{0}, and two distinct vertices x and y are adjacent if and only if ann R (xy) ≠ ann R (x) ∪ ann R (y). It follows that each edge (path) of the zero-divisor graph Γ(R) is an edge (path) of AG(R). In this article, we study the graph AG(R). For a commutative ring R, we show that AG(R) is connected with diameter at most two and with girth at most four provided that AG(R) has a cycle. Among other things, for a reduced commutative ring R, we show that the annihilator graph AG(R) is identical to the zero-divisor graph Γ(R) if and only if R has exactly two minimal prime ideals.
Skew Derivations and Engel ConditionsLanski, Charles
doi: 10.1080/00927872.2012.707719pmid: N/A
It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x k 0 ), x k 1 ], x k 2 ],…], x k n ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result in two variables is obtained for d a (θ, ϕ)-derivation.