symmetric monoidal (∞,1)-category of spectra
For $C$ a cartesian monoidal category (a category with finite products), an internal ring or a ring object in $C$ is an internalization to the category $C$ of the notion of a ring.
This is a monoid object internal to the category of abelian group objects internal to $C$.
Ring objects can be defined in more general symmetric monoidal categories as the corresponding module over a ring operad.
A ring object in Top is a topological ring.
A topos equipped with a ring object is called a ringed topos, see there for more details.
ring, ring object