Can an element in a commutative Noetherian ring have factorizations of arbitrary length?  Can there be an element $r$ such that for each

0$, there is a factorization $r=a_{1n}a_{2n}\cdots a_{nn}$ in which

all the factors are non-units?  If this isn't possible, suppose the
weaker hypothesis of ascending chain condition on principal ideals?

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