  We introduce the notion of ideal mutations in a triangulated category, which
generalizes the version of Iyama and Yoshino \cite{iyama2008mutation} by
replacing approximations by objects of a subcategory with approximations by
morphisms of an ideal. As applications, for a Hom-finite Krull-Schmidt
triangulated category $\mathcal{T}$ over an algebraically closed field $K$. (1)
We generalize a theorem of Jorgensen \cite[Theorem 3.3]{jorgensen2010quotients}
to a more general setting; (2) We provide a method to detect whether
$\mathcal{T}$ has Auslander-Reiten triangles or not by checking the necessary
and sufficient conditions on its Jacobson radical $\mathcal{J}$: (i)
$\mathcal{J}$ is functorially finite, (ii) Gh$_{\mathcal{J}}= {\rm
CoGh}_{\mathcal{J}}$, and (iii) Gh$_{\mathcal{J}}$-source maps coincide with
Gh$_{\mathcal{J}}$-sink maps; (3) We generalize the classical Auslander-Reiten
theory by using ideal mutations.
