In a paper by Sweedler (https://www.jstor.org/stable/1998053?seq=1#page_scan_tab_contents), he gives the following definition of a coring
Suppose $f:R\rightarrow S$ is a map of rings. $S$ need not be an $R$ algebra since $R$ may not be commutative. Even if $R$ is commutative it may not have central image in $S$. Nevertheless the ring structure on $S$ can be expressed in terms of two maps $$\psi:S\otimes_RS\rightarrow S:s_1\otimes s_2\mapsto s_1s_2$$ $$\phi:R\rightarrow S$$, which satisfy certain commutative diagrams. Reversing all the arrows leads to the notion of an $R$-coring.
Questions:
- I assume that when he says map of rings he means a ring morphism
- What is he trying to say? That given a ring $R$ and a map from $R$ to some set $S$, then saying that $S$ is a ring extension of $R$ is equivalent to some commutative diagrams with the maps he defines? If that is indeed what he is trying to show, how do we get what we want on our conditions on addition for $S$