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Abstract Algebra

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Integral Domains: Why is $Z/12Z$ not an Integral Domain?

ABSALG-WPN6D9

Let $R$ be a nonzero commutative ring with identity $1_R$.

Which of the following properties can $R$ share with $\mathbb{Z}/12\mathbb{Z}$ and still be an integral domain?

A

There exist $r,s\in R\setminus\{0\}$ with $rs=0$.

B

$R$ is isomorphic to a product of $2$ nonzero rings.

C

The set of units of $R$ is strictly contained in $R\setminus\{0\}$.

D

The characteristic of $R$ is not a prime number.