Wiki in https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication given a curve, $E$, defined along some equation in a finite field (such as $E: y^2 = x^3 + ax + b$), point multiplication is defined as the repeated addition of a point along that curve. Denote as $nP = P + P + P \dots + P$ for some scalar (integer) n and a point $P = (x, y)$ that lies on the curve, $E$. This type of curve is known as a Weierstrass curve.
My query is can we define something like this over elliptic curves over general rings? Also could multiplication be defined on non-Weierstrass curves?