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Abstract Algebra

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Unique Factorization Domain (UFD): General Definition & Examples

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Let $R$ be a commutative ring with identity $1_R$ which is an Integral Domain.

Then, by definition, $R$ is a Unique Factorization Domain (UFD) when:

(i) Every $r\in R$, $r\not=0$, that is not a unit in $R$ can be written as a product $r=s_1s_2\ldots s_k$ with $s_i\in R$ irreducible.
(ii) If $s_1s_2\ldots s_k=t_1t_2\ldots t_\ell$ with all $s_i, t_j\in R$ irreducible, then $k=\ell$ and we can reorder the $t_j$ so that $s_i=ut_i$ for a unit $u_i$ in $R$, $i=1,\ldots k$

Which of the following rings is not a UFD?

A

The ring of integers $\mathbb{Z}$

B

The ring $\mathbb{Z}+\mathbb{Z}\sqrt{-1}$

C

The ring $\mathbb{Z}+\mathbb{Z}\sqrt{-11}$

D

The ring of rational numbers $\mathbb{Q}$