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A commutative ring $R$ with unity is Hermite if for all $x,y\in R$ there exists $t,u,v\in R$ such that $x=tu$, $y=tv$ and $(u,v)=(1)$. Is there a finite commutative ring with unity that is not Hermite?

This characterisation is taken from theorem 3 of:

Some Remarks About Elementary Divisor Rings, Leonard Gillman and Melvin Henriksen, Transactions of the American Mathematical Society, Vol. 82, No. 2 (Jul., 1956), pp. 362-365

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+1: interesting question. But -1 for using "cru" to stand for "commutative ring with unity". Note that you have plenty of room to write this out. I will "remove my downvote" when this cru business is remedied. – Pete L. Clark Dec 19 '10 at 7:45
For lazy people like me... – J. M. Dec 20 '10 at 14:09
up vote 3 down vote accepted

Yes, $\mathbb{F}_2[x,y]/(x,y)^2$.

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Thanks! I'd like to use this observation in a paper - may I acknowledge you? If you'd like me to use your real name, send me an e-mail, cmcq of-email-domain – Colin McQuillan Nov 18 '10 at 8:42
Since there are not many caracters, you can use it free of charge;) – Plop Nov 19 '10 at 0:13

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