A semigroup S is called collapsing if there exists a positive integer n such that for every subset of n elements in S, at least two distinct words of length n on these letters are equal in S. In particular, S is collapsing whenever it satisfies a law. Let [Uscr ](A) denote the group of units of a unitary associative algebra A over a field k of characteristic zero. If A is generated by its nilpotent elements, then the following conditions are equivalent: [Uscr ](A) is collapsing; [Uscr ](A) satisfies some semigroup law; [Uscr ](A) satisfies the Engel condition; [Uscr ](A) is nilpotent; A is nilpotent when considered as a Lie algebra.