This accepted answer to the question about $BU_\otimes$ made me recall that I want to read about this general phenomenon for a long time.

What will follow is sort of vernacular but whether it can be translated into decent language is actually part of my question :D

So the double quoted words are all wrong but believed by me to be replaceable by something legitimate, and among other things I am asking whether this is the case and what the correct words must be.

So let us intuitively look at ring spectra as "actual" "spaces" with addition, multiplication and unit element satisfying ring identities "up to higher homotopies".

Such a guy $R$ has $R_0$ - the "connected component of 0" and $R_1$ - the "connected component of 1".

I know two examples when there is an "exponential map" $R_0\to R_1$ which is moreover an "isomorphism" between the additive structure of $R_0$ and multiplicative structure of $R_1$. Their "infinite loop space representatives" are $\Omega^\infty S^\infty$ and $\mathbb Z\times BU$ respectively, so although existence of these maps in these cases is seemingly very deep and important fact, in a sense it is trivial - by definition all connected components are just the same (well not actually trivial in the first case - it only becomes as trivial as the second after applying the highly nontrivial Barratt-Priddy equivalence $\Omega^\infty S^\infty\simeq\mathbb Z\times B\Sigma_\infty^+$).

Now the main part of my question.

Where can I read about a systematic study of such exponential maps (not necessarily isomorphisms) for various ring spectra?

**Later** -
as Charles Rezk says in the comment below, it does not make any sense to call this map exponential since it does not transform addition into multiplication at all.

I am still not sure whether I should withdraw the question altogether or in fact I still had to ask something sensible after all. Let me think some more...