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 $R$

please help me.

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$.

but I can't prove this or take counterexample .

  • What are your thoughts on the problem? For instance, what do you know about irreducibility? And take a look at the contrapositive statement too, because that might be easier to prove. – Arthur Aug 10 at 6:25
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    If you could "prove and disprove" it you would have proven all of mathematics inconsistent. Assuming you meant "or", please tell us something about your thoughts/work on the problem so we can offer help where the problem is and not just serve the complete solution. – Henrik Aug 10 at 6:30
  • Thank you for advise! – hew Aug 10 at 6:45
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    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

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