Say I have a ring $R$ which is not necessarily unital then am I correct in saying that the set $RR$ is a subset of $R$ and not necessarily equal to $R$?
Where the product of subsets $A$ and $B$ of a ring $R$ is defined to be the set of all finite sums of elements in the block product of $A$ and $B$.
I ask this because if $r \in R$ and there is no unital element then does $r$ necesscarily belong to the block product of $R$ with itself?
On
Of course.
Let $R$ be a ring with more than one element in which products are all zero.
Then $RR=\{0\}\neq R$.
Requiring $RR=R$ is a sort of regularity condition that can take the place of a missing identity. For example, the "infinite direct sum" of rings $R=\oplus F_2$ does not have an identity, but $RR=R$ (since $e^2=e$ for every element.)
No it does not. Look for example at the non-unital ring $R=2\mathbb{Z}$. Then $2 \notin RR$.