In ring theory, what does $R^{2} \neq \{0\}$ mean?

Question

I'm working on an exercise of Malik's Fundamentals of Abstract Algebra, namely: "Let $R$ be a ring such that $R^{2} \neq \{0\}$. Prove that $R$ is a division ring if and only if $R$ has no nontrivial left ideals."

The only thing I don't get is the meaning (and use) of $R^{2} \neq \{0\}$.

Answer 1

$R^2$ denotes the subset of $R$ containing all products of two elements. If this subset is equal to the set containing $0$, this means that multiplication of two elements in the ring always yields 0, so the multiplication is trivial.

Answer 2

Most probably $R^2=\{ab: a,b\in R\}$. Then $R^2\neq \{0\}$ implies that there are two elements of $R$ whose product is not $0$.