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Here is the last problem of my final exam in "Commutative algebra" which I think, no one has solved it completely, today!

Let $R$ be a (commutative with $1$) Noetherian ring. Let $a_1,...,a_n$ and $b_1,...,b_n$ be two regular sequences in $R.$ Prove that there is a regular sequence $c_1,...,c_n$ s.t. for each $i, \; 1 \leq i \leq n, \; c_i \in (a_1,...,a_i) \cap (b_1,..,b_i).$

Note: I attempted to show that $c_i=a_ib_i$ is the desired one, but it seems that we can not do anything, when $i \geq 2.$

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Your attempt does not work for the sequences $(x,y)$ and $(y,x)$ in $k[x,y]$. – Mariano Suárez-Alvarez♦ Jun 20 '11 at 11:46
Unfortunately, I figured it out after the exam! – Ehsan M. Kermani Jun 20 '11 at 11:50
why don't you post what you figured out (to be the answer)? – amWhy Jun 20 '11 at 12:55
@amWhy: if I knew, I wouldn't ask. – Ehsan M. Kermani Jun 25 '11 at 14:05
I'm sorry if my comment seemed rather off-handed; I was simply asking you what you gave as an answer (even if you weren't sure about it, or guessed)...just to help us give hints/suggestions...Perhaps that's what you meant by your attempt in your post. I thought you were asking it after having encountered it on exam, and in response to you response immediately above my comment: "I figured it out after the exam!" – amWhy Jun 25 '11 at 14:10

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