Lin and Su classified $A\mathcal{T}$-algebras of real rank zero. This class includes all $A\mathbb{T}$-algebras of real rank zero as well as many $C^∗$-algebras which are not stably finite. An $A\mathcal{T}$-algebra often becomes an extension of an $A\mathbb{T}$-algebra by an $AF$-algebra. In this paper, we show that there is an essential extension of an $A\mathbb{T}$-algebra by an $AF$-algebra which is not an $A\mathcal{T}$-algebra. We describe a characterization of an extension 𝐸 of an $A\mathbb{T}$-algebra by an $AF$-algebra if 𝐸 is an $A\mathcal{T}$-algebra.
