# Question About Irreductibility of an element in a ring

I $A$ is a principal ideal domain and let N the set of non zero elements, that do not have an inverse and that cannot be decomposed in a product of irreductibles.

If $a \in N$

Let's create the set $I = aA$

I am trying to understand why a is not irreductible

(This is part of the proof of the fundamental theorem of arithmetics our professor gave us)

Is it just that if $a$ was irreductible then $a = a$ so it can be decomposed in a product of irreductibles?

• I don't really understand your question. If $a$ cannot be decomposed into a product of irreducible elements, then $a$ can be both, irreducible and reducible. In a noetherian ring every element can be decomposed into irreducible elements. – Paul K May 31 '16 at 5:20
• Let me change the question – aribaldi May 31 '16 at 5:22
• Principal ideal domains are factorial – Paul K May 31 '16 at 5:35
• @menag that's precisely what i'm trying to understand thanks though – aribaldi May 31 '16 at 5:36
• @aribaldi What is the relevance of $I$ in your question? Is it part of your attempt at a solution. If so, how did you plan to use $I$? – M. Vinay May 31 '16 at 5:42

if A a factorial domain , for $a\in A$, a not 0 and not invertible, we have equivalence :

a irreductible

a prime

aA irreductible ideal

aA maximal ideal in the set of principal ideal and propre

aA prime ideal

• But I'm asking about the element a that cannot be decomposed in a product of irreductibles, why is that very element not irreductible? – aribaldi May 31 '16 at 9:06
• I tried to translate your question but I do not understand the problem. – m.idaya May 31 '16 at 9:27
• is that N is the subset of A containing all the irreducible elements of A? – m.idaya May 31 '16 at 9:33
• from the comments, i understant that you want to show a principal ring is factrial, is not it? – m.idaya May 31 '16 at 9:42
• i want to know why an element that cannot be decomposed into irreductibles cannot be an irreductible – aribaldi May 31 '16 at 12:22