Name for a ring that also has composition - aka function application?

This appears to be very similar to a composition ring, except that the axioms on Wikipedia don't include the axioms $0\circ c=0$ and $1\circ c=1$ (and indeed, don't require an unital ring, so there may not be an $1$), but require a commutative ring, which you don't require.

However, $0\circ c=0$ follows from $0\circ c = (0+0)\circ c = (0\circ c) + (0\circ c)$, so at least that axiom is superfluous.

For $1\circ c$ this doesn't work, since in a ring you don't need to have multiplicative cancellation. So this is a true extra condition to a composition ring; indeed, the linked Wikipedia üage contains an example that explicitly contradicts this axiom (namely, $f\circ g=0$ for all $f,g\in R$).