The book Primes of the form $x^2+ny^2$ by David A. Cox contains the following definitions regarding an elliptic curve (by which he means an equation $y^2=4x^3-g_2x-g_3$ such that $\Delta=g_2^3-27g_3^2$ is invertible) over a general (commutative, unitary) ring $R$ in which $2$ and $3$ are invertible:
Given an elliptic curve $E$ over $R$, we set $$ E_0(R) = \{(x,y)\in R\times R:y^2=4x^3-g_2x-g_3\}\cup\{\infty\} \text. $$ The reason for the new notation is that $E_0(R)$ may fail to be a group! [...] Using tools from algebraic geometry, one can define a superset $E(R)$ of $E_0(R)$ which is a group, but we prefer to use $E_0(R)$ because it is easier to work with in practice.
(Emphasis mine.) My question is: What are these "tools from algebraic geometry" and how does one apply them to this problem?