Let $𝒫$ be an arbitrary parabolic subalgebra of a simple associative $F$-algebra. The ideals of $𝒫$ are determined completely; Each ideal of $𝒫$ is shown to be generated by one element; Every non-linear invertible map on $𝒫$ that preserves ideals is described in an explicit formula.

In this paper we prove that every bijection preserving Lie products from a triangular algebra onto a normal triangular algebra is additive modulo centre. As an application, we described the form of bijections preserving Lie products on nest algebras and block upper triangular matrix algebras.

We shall introduce the class of strongly cancellative multiplicative monoids which contains the class of all totally ordered cancellative monoids and it is contained in the class of all cancellative monoids. If $G$ is a strongly cancellative monoid such that $hG\subseteq Gh$ for each $h\in G$ and if $R$ is a ring such that $aR\subseteq Ra$ for each $a\in R$, then the class of all non-singular left $R$-modules is a cover class if and only if the class of all non-singular left $RG$-modules is a cover class. These two conditions are also equivalent whenever...

Let $G$ be a multiplicative monoid. If $RG$ is a non-singular ring such that the class of all non-singular $RG$-modules is a cover class, then the class of all non-singular $R$-modules is a cover class. These two conditions are equivalent whenever $G$ is a well-ordered cancellative monoid such that for all elements $g,h\in G$ with $g<h$ there is $l\in G$ such that $lg=h$. For a totally ordered cancellative monoid the equalities $Z\left(RG\right)=Z\left(R\right)G$ and $\sigma \left(RG\right)=\sigma \left(R\right)G$ hold, $\sigma$ being Goldie’s torsion theory.

One of the results in my previous paper On torsionfree classes which are not precover classes, preprint, Corollary 3, states that for every hereditary torsion theory $\tau$ for the category $R$-mod with $\tau \ge \sigma$, $\sigma$ being Goldie’s torsion theory, the class of all $\tau$-torsionfree modules forms a (pre)cover class if and only if $\tau$ is of finite type. The purpose of this note is to show that all members of the countable set $𝔐=\{R,R/\sigma \left(R\right),R[x_1,\cdots ,x_n],R[x_1,\cdots ,x_n]/\sigma \left(R[x_1,\cdots ,x_n]\right),n<\omega \}$ of rings have the property that the class of all non-singular left modules forms a (pre)cover...

Let C n(A,B) be the relative Hochschild bar resolution groups of a subring B ⊆ A. The subring pair has right depth 2n if C n+1(A,B) is isomorphic to a direct summand of a multiple of C n(A,B) as A-B-bimodules; depth 2n + 1 if the same condition holds only as B-B-bimodules. It is then natural to ask what is defined if this same condition should hold as A-A-bimodules, the so-called H-depth 2n − 1 condition. In particular, the H-depth 1 condition coincides with A being an H-separable extension of B....

In this paper rings for which every $s$-torsion quasi-injective module is weakly $s$-divisible for a hereditary preradical $s$ are characterized in terms of the properties of the corresponding lattice of the (hereditary) preradicals. In case of a stable torsion theory these rings coincide with $TQI$-rings investigated by J. Ahsan and E. Enochs in [1]. Our aim was to generalize some results concerning $QI$-rings obtained by J.S. Golan and S.R. L’opez-Permouth in [12]. A characterization of the $QI$-property in the...