I'm trying to understand a little detail in this proof:
I didn't understand why in a ring we can always write an element as a products of non-units elements.
I need help.
Thanks in advance
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The question has been answered by the comments. In the sentence under question, the writer is dealing with the case where $a$ is reducible, so by definition of reducible, $a = bc$ for some nonunits $b$ and $c$.
It is untrue that all ring elements can be expressed as products of nonunits. The product $a = bc$ is a unit if and only if $b$ and $c$ are units.