Let R be an associative ring with 1 and R-tors the somplete Brouwerian lattice of all hereditary torsion theories on the category of left R-modules. A well known result asserts that R is a left semiartinian ring iff R-tors is a complete atomic Boolean lattice. In this note we prove that if L is a complete atomic Boolean lattice then there exists a left semiartinian ring R such that L is lattice-isomorphic to R-tors.

A ring A is called a chain ring if it is a local, both sided artinian, principal ideal ring. Let R be a commutative chain ring. Let A be a faithful R-algebra which is a chain ring such that Ā = A/J(A) is a separable field extension of R̅ = R/J(R). It follows from a recent result by Alkhamees and Singh that A has a commutative R-subalgebra R₀ which is a chain ring such that A = R₀ + J(A) and R₀ ∩ J(A) = J(R₀) = J(R)R₀. The structure of A in terms of a skew polynomial ring over R₀ is determined.

In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let $J^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very $J^{\#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of $J^{\#}$. A ring $R$ is said to be very $J^{\#}$-clean in case every element in $R$ is very $J^{\#}$-clean. We prove that every very $J^{\#}$-clean ring is strongly $\pi$-rad clean and has stable range one. It is shown...

This article presents a brief survey of the work done on rings generated by their units.

A ring R is called Armendariz (resp., Armendariz of power series type) if, whenever $\left({\sum }_{i\ge 0}{a}_i{x}^i\right)\left({\sum }_{j\ge 0}{b}_j{x}^j\right)=0$ in R[x] (resp., in R[[x]]), then $a_i{b}_j=0$ for all i and j. This paper deals with a unified generalization of the two concepts (see Definition 2). Some known results on Armendariz rings are extended to this more general situation and new results are obtained as consequences. For instance, it is proved that a ring R is Armendariz of power series type iff the same is true of R[[x]]. For an injective endomorphism σ of a ring...

Let R be a parabolic subgroup in $G{L}_n$. It acts on its unipotent radical $R_u$ and on any unipotent normal subgroup U via conjugation. Let Λ be the path algebra $k_t$ of a directed Dynkin quiver of type with t vertices and B a subbimodule of the radical of Λ viewed as a Λ-bimodule. Each parabolic subgroup R is the group of automorphisms of an algebra Λ(d), which is Morita equivalent to Λ. The action of R on U can be described using matrices over the bimodule B. The advantage of this description is that each...

We connect the theorems of Rentschler [rR68] and Dixmier [jD68] on locally nilpotent derivations and automorphisms of the polynomial ring $A_0$ and of the Weyl algebra $A_1$, both over a field of characteristic zero, by establishing the same type of results for the family of algebras $$A_h=\langle x,y\mid yx-xy=h\left(x\right)\rangle ,$$ where $h$ is an arbitrary polynomial in $x$. In the second part of the paper we consider a field $𝔽$ of prime characteristic and study $𝔽\left[t\right]$comodule algebra structures on $A_h$. We also compute the Makar-Limanov invariant of absolute constants...

Additive deformations of bialgebras in the sense of J. Wirth [PhD thesis, Université Paris VI, 2002], i.e. deformations of the multiplication map fulfilling a certain compatibility condition with respect to the coalgebra structure, can be generalized to braided bialgebras. The theorems for additive deformations of Hopf algebras can also be carried over to that case. We consider *-structures and prove a general Schoenberg correspondence in this context. Finally we give some examples.