Let A be a simple, σ-unital, non-unital, non-elementary C*-algebra and let I min be the intersection of all the ideals of M(A) that properly contain A. I min coincides with the ideal defined by Lin [19] in terms of approximate units of A and I min /A is purely infinite and simple. If A is separable, or if A has the (SP) property and its dimension semigroup D(A) of Murray-von Neumann equivalence classes of projections of A is order separable, or if A has strict comparison of positive elements by traces, then A = I min .If the tracial simplex T (A) is nonempty, let Icont be the closure of the linear span of the elements A ∈ M(A) + such that the evaluation map Â(τ ) = τ (A) is continuous. If A has strict comparison of positive element by traces then I min = Icont. Furthermore, I min too has strict comparison of positive elements in the sense that if A, B ∈ (I min ) + , B ∈ A and dτ (A) < dτ (B) for all τ ∈ T (A) for which dτ (B) < ∞, then A B.However if A does not have strict comparison of positive elements by traces then I min = Icont can occur: a counterexample is provided by Villadsen's AH algebras without slow dimension growth. If the dimension growth is flat, Icont is the largest proper ideal of M(A).