If $R$ and $S$ are two graded rings, is there a name for the construction $\oplus_{n \geq 0} R_n \otimes_{\mathbb{Z}} S_n$?
This gives a graded ring, but it is not quite the tensor product since we are only allowing tensors between two elements of the same degree.
This comes up when twisting a sheaf of quasi-coherent graded algebras $A$ on a scheme by a line bundle $L$ : $A' = \oplus_{n \geq 0} A_n \otimes L^{\otimes n}$. (Ravi exercise 17.2.G)
I am wondering if this object has a standard name, mostly so that I can look up properties about it to sanity check if necessary.