Ring on an Elliptic Curve

Question

I know that for a given elliptic curve $E$ we can define a group $G$ with the points on this curve. However, can we define a ring on it? That is, can we define a multiplication on the curve, where we take two points $P$ and $Q$ and produce another point $R$?

Note: I am not talking about the point multiplication, where a point $P$ is added to itself repeatedly.

Answer 1

Elliptic curves are used in cryptography because they do not have ring structures. See reference [12] in this paper.