In this thesis we study infinite-dimensional Lie algebras, drawing inspiration from group theory and ring theory. Chapter one sets up notation. Chapter two deals with prime ideals. In the first part of it, we define the concepts of a prime ideal and the radical of an ideal in Lie algebras along the same line as ideals in an associative rings, and investigate some of their properties. [...] In the second we investigate the structure of Lie algebras with certain finiteness conditions on subideals, using the notion of prime ideals and prime algebras. [...] Chapter three deals with generalizations of the minimal condition on ideals, leading to a new class of "quasi-Artinian" algebras (We say that L is quasi-Artinian if for every descending chain of ideals [...] Chapter four considers the join of subideals. [...] This result is a counterpart of a group- theoretic one (cf. Wielandt [35]). We also find another condition under which the join of subideals is a subideal by imposing conditions on the circle product H°K = [H,K] [...] Chapter five considers criteria for subideality and ascendancy generalizing some results of Kawamoto [17], Stitzinger [28]. [...] Finally if L is an ideally finite Lie algebra over a field of characteristic zero and if H < l [...]