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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.2010-12-19
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    [For lazy people like me...](http://www.ams.org/journals/tran/1956-082-02/S0002-9947-1956-0078979-8/S0002-9947-1956-0078979-8.pdf)2010-12-20

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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 liv.ac.uk.2010-11-18
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    Since there are not many caracters, you can use it free of charge;)2010-11-19