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

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4 Answers 4

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In a finite commutative ring with unity, every element is either a unit or a zero-divisor. Indeed, let $a\in R$ and consider the map on $R$ given by $x \mapsto ax$. If this map is injective then it has to be surjective, because $R$ is finite. Hence, $1=ax$ for some $x\in R$ and $a$ is a unit. If the map is not injective then there are $u,v\in R$, with $u\ne v$, such that $au=av$. But then $a(u-v)=0$ and $u-v\ne0$ and so $a$ is a zero divisor.

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+1: My answer looks like overkill compared to this. –  Pete L. Clark Aug 31 '11 at 16:25
    
If you want to know when the converse holds, see mathoverflow.net/questions/42647/…, which is related to Pete's answer. –  lhf Aug 31 '11 at 16:29
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@Pete: Most of your answers are overkill, which is why I enjoy them so much! :-) –  Asaf Karagila Aug 31 '11 at 16:47
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Perhaps my answer can be better phrased as: If the map is surjective then $a$ is a unit. Otherwise, the map cannot be injective, because $R$ is finite, and so $a$ is a zero divisor. –  lhf Aug 31 '11 at 18:32
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I think for a complete solution, you still have to state that no element can be both a unit and a zero divisor. That's not a big deal, of course. –  azimut Sep 13 '13 at 10:50

Your question is incomplete: you say you want to prove that every nonzero element of $R$ is "either a zero-divisor?" If one inserts a unit or before zero-divisor then you get a true statement, so I'll assume for now that's what you meant.

First, following a comment by Gerry Myerson on a recent related answer, let me divulge that for me zero is a zero-divisor. I claim that this is just a convention that you should be able to translate back from if you see fit.

Next, note that if you have a family $\{R_i\}_{i \in I}$ of rings in which every element is either a unit or a zero-divisor, the same holds in the Cartesian product $R = \prod_{i \in I} R_i$.

In your case you can use the structure theorem for Artinian rings: $R$ is a finite product of local Artinian rings -- to reduce to the case in which $R$ is local Artinian. Then the maximal ideal is nilpotent, so every nonunit is nilpotent and in particular a zero divisor.

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+1: Funny! It's like killing birds with cannon balls. :D I had never thought that a finite ring is Artinian. –  Andrea Aug 31 '11 at 18:36
    
@Andrea Another application of $\rm\:R\:$ finite $\rm\:\Rightarrow\: R\:$ Artinian is in this prior answer: A finite ring is a field if its units $\cup\ \{0\}$ comprise a field of characteristic $\ne 2$. –  Bill Dubuque Sep 3 '11 at 23:00

HINT $\rm\ |R|<\infty\ \Rightarrow\ r^j=r^k,\: j<k\ \Rightarrow\ r^j\:(1-r^{k-j})=0\ \Rightarrow\ 1 = r^{k-j}\: $ if $\rm\:r\:$ not a zero-divisor.

NOTE $\ $ The idea generalizes: if a non-zero-divisor $\rm\:r\:$ is algebraic then it divides the least degree coefficient of any polynomial of which it is a root. When said coefficient is a unit then so too is $\rm\:r\:.\:$ Hence the result holds more generally for any ring satisfying a polynomial identity whose least degree coefficient is unit, e.g. for Jacobson's famous rings satisfying the identity $\rm\:X^n =\: X\:.$

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Since one good cannonball deserves another, I'd like to provide a solution using right Artinian rings that aren't necessarily commutative.

Definitions:

A ring $R$ is called strongly $\pi$-regular if for all $x\in R$, chains of the form $xR\supseteq x^2R\supseteq x^3R\supseteq\dots \supseteq x^iR\supseteq\dots$ become stationary.

A ring is called Dedekind finite if $xy=1$ implies $yx=1$ for all $x,y\in R$.

Strongly $\pi$-regular rings were introduced by Kaplansky in the citation at the bottom. The definition is usually given in terms of "$\forall x\exists r(x^n=x^{n+1}r)$", but this is equivalent.

Moreover, it's been shown that $r$ can be chosen to commute with $x$, and so the left-hand version of this definition is equivalent to this one.

It's obvious right Artinian rings are strongly $\pi$-regular, and it turns out they are Dedekind finite too.

Proposition: In a strongly $\pi$-regular Dedekind finite ring (in particular, right or left Artinian rings), each element is a unit or a zero divisor. (Zero being counted as a zero divisor.)

Proof: Let $x\in R$ be a nonunit, and let $n$ be minimal such that there exists $r$ that commutes with $x$ and $x^n=x^{n+1}r$. Since $x$ isn't a unit, $n\geq 1$. (Because if $1=xr$, $x$ would be a unit by Dedekind finiteness.)

Rearranging, we get $x(x^{n-1}-x^nr)=0=(x^{n-1}-x^nr)x$ since $r$ commutes with $x$. By minimality of $n$, $x^{n-1}-x^nr\neq 0$. Thus, $x$ has been demonstrated to be a two-sided zero divisor.


I. Kaplansky, Topological representations of algebras II, Trans. Amer. Math. Soc. 68 (1950), 62-75. MR 11:317

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