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

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)(L) is Hopfian, then the c-center of every c-stem cover of L is mapped onto Z∗c(L).