Artinian Tor modules (Reference request)

I am looking for a reference for the following basic fact:

Let $$R$$ be a noetherian ring, let $$M$$ be an artinian $$R$$-module, let $$N$$ be a finitely generated $$R$$-module, and let $$i\in\mathbb{N}$$. Then, $$Tor_i^R(M,N)$$ is artinian.

(I know that it is easy to prove. I guess nevertheless that this is written down somewhere in the standard literature about homological algebra, and I would like to know where.)

Remark 1: The above conclusion also holds if $$R$$ is coherent, $$M$$ is artinian, and $$N$$ is of finite presentation. A reference for this generalisation would also be fine.

Remark 2: Leamer proves in his PhD Thesis (2010) a generalisation to the case where $$R$$ is noetherian, $$M$$ is artinian, and $$N$$ is minimax. But I guess there must be an earlier reference for the less general case.

• I doubt that there is a precise reference. As you point out, this is a straightforward consequence of the definitions; if I had to use it, I'd just say that, I wouldn't bother to write a lemma. – abx Nov 24 '19 at 13:05
• Ok, it seems there is no such reference. Thanks, @abx, for your suggestion, to which I will probably adhere. – Fred Rohrer Nov 27 '19 at 12:26