# Can every element of a Noetherian ring be expressed as a product of irreducible elements?

Let $X$ be a Noetherian ring.

Question: Can every element of $X$ be expressed as product of irreducible elements?

I'm not assuming that $X$ is a Noetherian integral domain, only $X$ is Notherian.

The Question is hold if $X$ is Noetherian.

Can anyone give me a suggestion?

• math.stackexchange.com/questions/714719/… – Tsemo Aristide Mar 28 '16 at 17:35
• I'm not assuming that X is a Notherian integral domain, only X is Notherian. – Phalton Mar 28 '16 at 18:14
• @TsemoAristide: The OP does not assume $X$ to be an integral domain, an assumpation made in the question that you have pointed to. – Alex M. Mar 28 '16 at 18:33

I don't know what you mean by an irreducible element, but if this means a non-zero non-invertible element which can't be written as a product of two non-invertible elements, then for $X=\mathbb Z/6\mathbb Z$ the property fails for the simple reason that $X$ has no irreducible elements.