Let $R\neq \{0\}$ be a non-unital ring, where the multiplication is not everywhere zero, meaning there exist some nonzero elements $a,b\in R$ such that $ab\neq 0$.
Question 1: If $a\in R\setminus \{0\}$ is such that its left annihilator is the entire ring: $$\text{Ann}_R(a):=\left\{r\in R: ra=0\right\}=R$$ Can we conclude $a=0$?
Question 2: Is the answer the same for non-unital Banach algebras?
The difficulty here is we don't have invertible elements, as there is no identity element in the ring/algebra.
The condition you ask for is stronger than being a left zero divisor. I would prefer to call it something else. How about left annihilator (of $R$)?
For another example then, take a non-principal ideal $I$ of a ring $R$ with $1\not \in I$, and an element $x\in I$. Then consider $R'=I/(x)I$. E.g. $(x,y)/(x^2,xy)$ in $\mathbb{Z}[x,y]$.