Suppose we have the ring $\Bbb Z[x]$. I tried to find that the ring $\Bbb Z[x]/\langle x^2-1\rangle$ is isomorphic to $\Bbb Z\times\Bbb Z$ or not??? I tried to apply the Chinese Remainder theorem. For it I required that ideals $I=\rangle x+1\langle$ and ideal $J$ generated by $x-1$ must be comaximal. But unable to get anything. Please anyone tell is above two rings are isomorphic or not??
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1The ring is not isomorphic to $\mathbb{Z}\times \mathbb{Z}$. I think you are a bit confused. – Bombyx mori Oct 20 '20 at 18:58
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@Bombyxmori please explain how we show that they are not isomorpic – Shubham singla Oct 20 '20 at 19:01
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At the duplicates, you can see that the idea of using $(x+1)$ and $(x-1)$ are NOT comaximal. Their sum is $(2, x+1)$, which is maximal. – rschwieb Oct 20 '20 at 19:04
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@Shubhamsingla The duplicates have multiple explanations as to why they are not isomorphic. The simplest one IMHO is the one that points out the quotient has two idempotents, but $\mathbb Z\times \mathbb Z$ has four idempotents. Are you bothering to read anything here? – rschwieb Oct 20 '20 at 19:05
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@rschwieb please help how to show that quotient ring has two idopotent elements... – Shubham singla Oct 20 '20 at 19:14
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Ie has only tribal idopotent elements – Shubham singla Oct 20 '20 at 19:14
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@rschwieb please give link of its dublicate answer – Shubham singla Oct 20 '20 at 19:25
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@Shubhamsingla It IS linked. See the colored box at the top of this post. – rschwieb Oct 20 '20 at 19:28
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@rschwieb but there is nothing about the co maximal ideal as you provide Meir I+J= <2,x+1> – Shubham singla Oct 20 '20 at 19:30
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@Shubhamsingla Obviously $x-1$ and $x+1$ are in $(x+1, 2)$, so the sum of their principal ideals is contained in that ideal. The reverse containment is equally clear. If you really need another link https://math.stackexchange.com/a/2251575/29335 – rschwieb Oct 20 '20 at 20:00
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I added an answer in the linked dupe which makes it much clearer conceptually why no ring factorization exists when comaximality fails. – Bill Dubuque Oct 20 '20 at 22:26