This paper concerns the role of divisors in abstract ideal theory. The setup is an algebraic multiplicative lattice \(\mathcal A\) (AML) by which here is meant a compactly generated (by a fixed set \({\mathcal A}_c\) of compact elements including 0 and 1) complete lattice with a not necessarily commutative multiplication which is distributive over arbitrary joins and usually has greatest element 1 as a compact multiplicative identity. For elements \(A,B\in L\), \(A\) is a left divisor of \(B\) if \(B = AX\) for some \(X\in L\). A right divisor of \(A\) is defined similarly and \(A\) is a divisor of \(B\) if it is both a left and right divisor of \(A\). And \(A\) is called a left divisor if it left divides each \(B\leq A\). Finally \(\mathcal A\) is called a left divisor AML if each element of \({\mathcal A}_c\) is a left divisor and a left Prüfer AML if each compact element is a left divisor. Thus the lattice of ideals of a Prüfer domain is a Prüfer AML This paper mainly investigates Prüfer AML's and their relationship to the lattice of certain star-ideals (such as \(v\)-ideals and \(t\)-ideals) on monoids.