Pre-derivations and description of non-strongly nilpotent filiform Leibniz algebras

In this paper we give the description of some non-strongly nilpotent Leibniz algebras. We pay our attention to the subclass of nilpotent Leibniz algebras, which is called filiform. Note that the set of filiform Leibniz algebras of fixed dimension can be decomposed into three disjoint families. We describe the pre-derivations of filiform Leibniz algebras for the first and second families and determine those algebras in the first two classes of filiform Leibniz algebras that are non-strongly nilpotent.

A note on the generalized nilpotent elements of a module

In this paper, we introduce the notion of the generalized nilpotent element of a module. In \cite{Groenewald}, the notion of nilpotent element of a module is introduced in the following sense: a non-zero element $m$ of an $R$-module $M$ is said to be nilpotent if there exists some $a\in R$ such that $a^k m=0$ but $am\neq 0$ for some $k\in \mathbb N$. In our present work we aim to generalize this notion. We have extended this notion to the strongly nilpotent element of a module.

On radical formula in modules over noncommutative rings

This paper examines the radical formula in noncommutative case and for this purpose, a generalization of prime submodule is defined. It is proved that there is a direct connection between onesided prime ideals and one-sided prime submodules. Moreover the connections between the intersection of all one-sided prime submodules and strongly nilpotent elements of a module are studied.

On the Strong Nilpotence Problem

In this paper, we describe the structure of quadratic homogeneous polynomial maps F=X+H with JH3=0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F=X+H is linearly triangularizable.

A GENERALIZATION OF BAER'S LOWER NILRADICAL FOR MODULES

Let M be a left R-module. A proper submodule P of M is called classical prime if for all ideals [Formula: see text] and for all submodules N ⊆ M, [Formula: see text] implies that [Formula: see text] or [Formula: see text]. We generalize the Baer–McCoy radical (or classical prime radical) for a module [denoted by cl.rad R(M)] and Baer's lower nilradical for a module [denoted by Nil *(RM)]. For a module RM, cl.rad R(M) is defined to be the intersection of all classical prime submodules of M and Nil *(RM) is defined to be the set of all strongly nilpotent elements of M (defined later). It is shown that, for any projective R-module M, cl.rad R(M) = Nil *(RM) and, for any module M over a left Artinian ring R, cl.rad R(M) = Nil *(RM) = Rad (M) = Jac (R)M. In particular, if R is a commutative Noetherian domain with dim (R) ≤ 1, then for any module M, we have cl.rad R(M) = Nil *(RM). We show that over a left bounded prime left Goldie ring, the study of Baer–McCoy radicals of general modules reduces to that of torsion modules. Moreover, over an FBN prime ring R with dim (R) ≤ 1 (or over a commutative domain R with dim (R) ≤ 1), every semiprime submodule of any module is an intersection of classical prime submodules.