# Verify that if $u$ is irreducible in $Z_n$ , then $u$ is irreducible in $Z_n[x]$.

This looks so simple but .. I can't prove or disprove this.

[Definition (irreducible element of ring)] ☞ Let $R$ is commutative ring with unity $1$. $u$ is irreducible elements in $R$ iff $u$ is non-zero, non-unit, and $u = ab$ imply $a$ is unit or $b$ is unit of $R$

I already know $u$ is irreducible elements in $Z_n$ ring iff $u$ associate $p$( that is, $u=pc$ , $c$ is unit in $Z_n$) , where $p$ is prime and $p^2$divide $n$.
• The term "irreducible" is normally reserved for elements of integral domains. The only $\mathbb{Z}_n$ which are integral domains are $\mathbb{Z}_p$ for $p$ a prime, and in this case the ring is a field and the result is vacuously true ($\mathbb{Z}_p$ has no irreducible elements!). What is your definition of "irreducible" in a ring which isn't an integral domain? – lokodiz Aug 10 at 14:16
• definition is here! : $u$ is irreducible element of ring $R$ iff $u=ab$ imply $a$ or $b$ is unit element of R – hew Aug 10 at 16:44