So, I'm reading Rognes's paper Galois Extensions of Structured Ring Spectra and was wondering about one of his claims. In the beginning of section 4.2 he says the following:
The Eilenberg–Mac Lane functor $H$, which to a commutative ring $R$ associates a commutative $S$-algebra $HR$ with $\pi_*HR = R$ concentrated in degree 0, embeds the category of commutative rings into the category of commutative $S$ -algebras.
What does "embedding" mean in this context? That the functor $H$ is fully faithful? Although I can see that this functor is faithful, it is not obvious to me that it is full (if it is). Is the Eilenberg-Mac Lane functor a full functor between the category of commutative rings to the category of commutative $S$-algebras?
It seems to me that different authors use the notion of "embedding" differently, so I just want to make sure that I am not stuck trying to prove something that is not true...