TITLE: Is the tensorproduct of a triangulated category with a ring again triangulated?
QUESTION [7 upvotes]: $\underline{Background}$ : 
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.

REPLY [8 votes]: It seems unreasonable to expect tensoring the morphisms in a triangulated category by a ring to be a meaningful operation - after all these morphisms are typically cohomology groups of natural complexes, and we know there are universal-coefficient-theorem type corrections to tensoring cohomology with a ring as opposed to the more natural operation of tensoring a complex.
On the other hand, if we enrich our triangulated categories everything works fine.
For example on the level of (pretriangulated) differential graded categories or $A_\infty$ categories, or if you prefer, stable $\infty$-categories, there's a perfectly nice
operation of tensor product by a ring - in fact one can tensor by another such category (generalizing the case you're asking about, which is tensoring with R-mod). Even more, there's a structure of symmetric monoidal $\infty$-category on such categories (developed in Lurie's Derived Algebraic Geometry series (an exposition and references can be found eg here).