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absorbing element
An element $\zeta$ of a groupoid $(G,\,*)$ is called an absorbing element (in French un élément absorbant) for the operation “$*$”, if it satisfies
$\zeta\!*\!a\;=\;a\!*\!\zeta\;=\;\zeta$ 
for all elements $a$ of $G$.
Examples

The zero $0$ is the absorbing element for multiplication (or multiplicatively absorbing) in every ring $(R,\,+,\,\cdot)$.

The zero ideal $(0)$ is absorbing for ideal multiplication.

The zero vector $\vec{0}$ is the absorbing element for the vectoral multiplication “$\times$”.

The empty set $\varnothing$ is the absorbing element for the intersection operation “$\cap$” and also for the Cartesian product “$\times$”.

In an upper semilattice, an element is absorbing iff it is the top element. Dually, an element is absorbing iff it is the bottom element in a lower semilattice.
As the examples give reason to believe, the absorbing element for an operation is always unique. Indeed, if in addition to $\zeta$ we have in $G$ another absorbing element $\eta$, then we must have $\eta=\zeta\!*\!\eta=\zeta$.
Because $\zeta\!*\!\zeta=\zeta$, the absorbing element is idempotent.
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Also mentions by CWoo ✓
minor by CWoo ✓
absorbing elements in semilattices by CWoo ✓