Let $R$ be a ring without a unit element and $R'$ be a (non trivial) ring with a unit element. Can there be an onto homomorphism from $R$ to $R'$?
Some observations: There cannot be an isomorphism, because then the element of $R$ which maps to $1$ in $R'$ must be a unit element in $R$. Also, we cannot have an onto homomorphism from a ring with unity to a ring without unity as $f(1)$ is going to be a unit element.
Edit: I am defining a ring homomorphism as a function $ \phi: R \rightarrow R'$ such that for all $a,b$ in $R$, $$\phi(a+b) = \phi(a) + \phi(b) $$ $$\phi(ab) = \phi(a)\phi(b)$$