On the c-Covers and a Special Ideal of Lie Algebras

Abstract

In this paper, our aim is to deal with some properties of c-covers of Lie algebras whose c-nilpotent multipliers are Hopfian. Moreover, it is proved that all c-covers of any nilpotent Lie algebra have Hopfian property and give a sufficient condition for two c-covers of such Lie algebras to be isomorphic. Also, we introduce a special ideal, denoted by $Z_{c}^{*} (L)$ in every Lie algebra $L$, which is the intersection of special subalgebras, then give another form of this ideal and study the connection between this ideal and the concept of the c-nilpotent multiplier. Finally, we prove that if $L$ is a Lie algebra for which $M^{(c)} \left( L \right)$ is Hopfian, then the c-center of every c-stem cover of $L$ is mapped onto $Z_{c}^{*} (L)$.