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For a binary (commutative) operation $*$ over a set $S$ the identity element is an element $i\in S$ such that $$\bigl(\forall x\in\ S \bigr)\bigl(i*x=x\bigr)$$ Is it correct/common to call an element $a\in S$ such that $$\bigl(\forall x\in\ S \bigr)\bigl(a*x=a\bigr)$$ the null element?

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This is more commonly known as an absorbing element. The alternative name zero element is used when there is no confusion with some other kind of zero already present in the set.

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Well, the zero element $0$ of a ring $R$ is absorbing, i.e., $r0 = 0 =0r$, since $r0 = r(r+(-r)) = r^2 - r^2=0$. So maybe absorbing is the notion looked for here.

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