Specific elements of a ring

Question

I'm studying Ring Theory and I have 3 questions, if could be nice to hear your comments about that.

1) I would like to find a primer element which is not irreductible. We said that $a$ is irreductible iff $a \in A - (A^{\times} \cup \{0\})$ and $$ \forall (b,c) \in A, a = bc \Rightarrow b \in A^{\times} \text{ or } c \in A^{\times} $$ If the ring $R$ is a integral domain then primer => irreductible. We said that $a$ is a primer element iif $a \in A - (A^{\times} \cup \{0\})$ and $$ \forall b,c \in A, a|bc \Rightarrow a|b \text{ or } a|c $$

2) I'm trying to find the primer and the irreductible element of $\mathbb Z/n\mathbb Z$, but I can just study exemples when $n$ is not a prime. Do you know how to study it in general ?

Thanks for your help and regards.

Answer 1

Here is Μάρκος Καραμέρης's answer:

To find and example of a prime element that is not irreducible consider $\mathbb{Z}/6\mathbb{Z}$ and the prime element $2=2×4$.

Elements that you call irreducible will be omnipresent in $\mathbb{Z}/p\mathbb{Z}$ ($p$ a prime) since this is a field, and $\mathbb{Z}/p\mathbb{Z}-(\mathbb{Z}/p\mathbb{Z}^\times \cup \{0\} )=\varnothing$.