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In an algebraic system, an element that behaves like $0$ does for addition or like 1 does for multiplication of real numbers is called an " identity element" ; an element that is to some element $a$ like $\frac 12$ is to 2 is called an inverse element of $a$.

Is there a general name for an element that behaves like $0$ does for multiplication?

I mean, a general name for an element $n$ having this property : $ \forall(a) a * n = n*a=n$ , $*$ being a binary operation?

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    $\begingroup$ We have $0\cdot a=a\cdot 0=0$ for all $a\in R$. "0" is just the null-element of the ring R. $\endgroup$
    – Peter
    Mar 21 '20 at 16:18
  • $\begingroup$ If you are assuming a ring or a field the only such element is the additive identity and it is just called "$0$" (As by distribution $0\times a = (0+0)\times a = 0\times a + 0\times a$ so $0\times a$ must but equal to $0$. And it is unique as $n=n*a=n*(a+a)=n*a+n*a=n+n$ so $n=0$.) But for abstract structures that need not be rings or fields then .... Well, Ben Steffan answer has the word for it. $\endgroup$
    – fleablood
    Mar 21 '20 at 16:27
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    $\begingroup$ @Peter Well, to be fair the OP never stated we were working with a Ring. S/he wanted the name for such an element with any binary operation on a set. $\endgroup$
    – fleablood
    Mar 21 '20 at 16:33
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Yes, such an element is known as an absorbing element.

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  • $\begingroup$ the name is properly given! $\endgroup$
    – user655689
    Mar 21 '20 at 16:19
  • $\begingroup$ Please search first before posting yet more dupe answers to FAQs. $\endgroup$ Mar 21 '20 at 17:00