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.

Given a group G of k-linear automorphisms of a locally bounded k-category R, the problem of existence and construction of non-orbicular indecomposable R/G-modules is studied. For a suitable finite sequence B of G-atoms with a common stabilizer H, a representation embedding ${\Phi }^B:Iₙ-spr\left(H\right)\to mod(R/G)$, which yields large families of non-orbicular indecomposable R/G-modules, is constructed (Theorem 3.1). It is proved that if a G-atom B with infinite cyclic stabilizer admits a non-trivial left Kan extension B̃ with the same...

We prove a stronger form, $A^{+}$, of a consistency result, $A$, due to Eklof and Shelah. $A^{+}$ concerns extension properties of modules over non-left perfect rings. We also show (in ZFC) that $A$ does not hold for left perfect rings.

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...