# Zero divisor in ring without unity [closed]

Let $R$ be a commutative ring without unity and $n \in R\setminus\{0\}$. Prove that $n\mid n$ implies that $n$ is a zero divisor.

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• Is this a commutative ring? – Arthur Nov 3 '16 at 16:40
• Yes, it is. I forget to mention that. – Taufi Nov 3 '16 at 16:41
• We can prove it only for commutative ring. For the not commutative ring: Let n = kn nn = knn nn = nkn 0 = (kn - nk)*n => n is the zero divisor or kn = n*k and we can go to subring, created by n and k – kotomord Nov 3 '16 at 16:43
• Can we extend R to a ring with unity? – Jacob Wakem Nov 3 '16 at 17:33

Assume that $n=nk$ for some $k\in R$. Because $R$ has no unity, there exists an element $r\in R$ such that $kr\neq r$. In other words $kr-r\neq0$.

But by standard applications of rng axioms $$n(kr-r)=n(kr)-nr=(nk)r-nr=nr-nr=0.$$ Therefore $n$ is a zero-divisor.

• Thanks so much, Jyrki Lahtonen. You saw the trick! – Taufi Nov 3 '16 at 17:07
• For an example of a rng where this can happen consider the direct sum of infinitely many copies of $\Bbb{Z}$. In other words $$R=\{(n_1,n_2,\ldots)\mid n_i\in\Bbb{Z}\ \text{such that n_i\neq0 for only finitely many i}\}.$$ This is a rng with componentwise operations. Clearly $n=(1,0,0,0,\ldots)$ is divisible by itself. And also a zero divisor. – Jyrki Lahtonen Nov 3 '16 at 17:25
• Where is commutativity used? – Serge Seredenko Nov 3 '16 at 20:47
• @Serge I use commutativity in concluding that there exists an $r$ such that $kr\neq r$ specifically. Otherwise I would need to worry about the possibility that $k$ might be a one-sided unity but not a two-sided one. That is, I avoid the headache possibility that $kr=r$ for all $r$, but may be $rk\neq r$ for some $r$, when we still could not conclude that $k$ is a multiplicative neutral element. My exposure to rngs is kinda lacking, so I can't tell right away, whether it is possible that a rng could have a one-sided identity :-) – Jyrki Lahtonen Nov 3 '16 at 20:51
• @JyrkiLahtonen Moreover, if $R$ is not commutative, $n\mid n$ is not well defined: one should distinguishing between left divisor and right divisor. – egreg Nov 3 '16 at 21:48