The first example of a ring spectrum is probably the Eilenberg-McLane spectrum of a ring $R$. But how is the multiplication $\mu: \mathbf H R \wedge \mathbf H R \to \mathbf H R$ defined? Probably this is so trivial that no book includes an explanation, but I really can not figure this out.
I am using the basic definition of spectra as sequential, possibly CW-, spectra given by Adams, I have found details on the multiplication on Schwede's book on symmetric spectra, but I am still not comfortable with those.