Let $ X, Y$ , let’s say, homotopy commutative $ H$ -spaces, $ f,g$ maps from $ X$ to $ Y$ . (Actually we only need $ Y$ to be homotopy commutative $ H$ _space, but the statement is easier if we also suppose $ X$ to be one). It is not difficult to show that

$ Q(f+g)_*=Qf_* +Qg_*$ where $ Q$ denotes the module of indecomposables, and $ _*$ denotes, let’s say, induced map in mod $ p$ ordinary homology. (Actually any generalized homology works here as long as we have Kunneth isomorphism.)

This can be shown either by directly writing down the two sides (if we denote $ $ \Delta (x)= x \otimes 1 +1\otimes x +\Sigma x’ \otimes x”$ $ then, we have $ $ (f+g)_*(x)= f_*x +g_*x + \Sigma f_*x’ \otimes g_*x”\mbox{ })$ $ or by noting that both $ _*$ and Q are additive functors.

Now my question is: I have seen this written somewhere, could anyone tell me where I can find a published reference?

This is something completely trivial once you know. However,

1. it is not so well-known
2. if I hadn’t read it some where, then I probably wouldn’t have found it on my own
3. it makes the life a lot easier.

To illustrate these points, I can say that I have seen a paper where the author computes $ (f+g)_*$ in a great length then only to pass to the indecomposable quotients. So the author and the referee of the paper weren’t able to come up with this.

So I really would like to give credit to whoever wrote this fact in a published paper. Thank you in advance.