$\underline{Background}$ : Suppose $\tau$ is a preadditive category and $R$ a ring. Then one may form a new preadditive category $\tau \otimes R$ in the following way:

$\tau \otimes R$ has the same objects as $\tau$ and for objects $A$ and $B$, set $\tau \otimes R(A,B) := \tau(A,B)\otimes_{\mathbb{Z}} R$. Composition is given by $\tau(A,B)\otimes_{\mathbb{Z}} R\otimes_{\mathbb{Z}}\tau(B,C)\otimes_{\mathbb{Z}} R \cong \tau(A,B)\otimes_{\mathbb{Z}}\tau(B,C)\otimes_{\mathbb{Z}}R\otimes_{\mathbb{Z}}R \rightarrow \tau(A,C) \otimes_{\mathbb{Z}}R$, where the last arrow is given by the composition in $\tau$ and multiplication in $R$.

$\tau \otimes R$ is additve if $\tau$ is additive.

$\underline{Question}$ : Suppose $\tau$ is triangulated, is there a (canonical) structure of a triangulated category for $\tau \otimes R$?

Actually, I only need the case $R := \mathbb{Z}[\frac{1}{n}, \theta] \subset \mathbb{C}$, where $\theta$ is a primitive root of unity of order $n$ for some $n \in \mathbb{N}$, but I would also be interested in a general statement/counterexample to the general case.

Thank you for reading.