Every nonzero element in a finite ring is either a unit or a zero divisor

Let R be a finite ring with unity. Prove that every nonzero element of R is either a unit or a zero-divisor.


In a finite commutative ring with unity, every element is either a unit or a zero-divisor. Indeed, let aR and consider the map on R given by xax. If this map is injective then it has to be surjective, because R is finite. Hence, 1=ax for some xR and a is a unit. If the map is not injective then there are u,vR, with uv, such that au=av. But then a(uv)=0 and uv0 and so a is a zero divisor.

For the noncommutative case, see this answer.

Source : Link , Question Author : rupa , Answer Author : lhf

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