An element $a$ in a monoid $A$ such that $a\cdot x=a$ for any $x\in A$

Question

Let $(A,\cdot)$ be a non-abelian monoid. I was just wondering if there is a name or terminology for an element $a\in A$ that satisfies one of the following propertes:

(1) For any $x\in A$, we have $x\cdot a= a$.

Or

(2) For any $x\in A$, we have $a\cdot x=a$.

Answer 1

(1) is called a right zero, (2) is called a left zero. I'm pretty sure there's no general term for both, but you can just say 'left or right zero'.