When are Associates Unit Multiples?

Overview of attention for article published in Rocky Mountain Journal of Mathematics, September 2004

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Two generators of a left cyclic module over an Artinian ring must be unit multiples?

Posted by Yifeng Huang, at mathoverflow.net, 29 Apr 2025

I suspect the following statement is true but I am stuck on producing a proof: Claim. Let $R$ be an Artinian ring (in case it's…

Is there a simple proof that elements of $\mathbb{Z}_n$ are unit multiples of each other if and only if they have the same gcd with $n$?

Posted by Conifold, at math.stackexchange.com, 27 Jan 2024

That is $b=ua$ with $u\in\mathbb{Z}_n^*$ if and only if $(a,n)=(b,n)$. It is easy enough to prove the forward direction using…

In arbitrary commutative rings, what is the accepted definition of "associates"?

Posted by goblin GONE, at math.stackexchange.com, 10 Jul 2014

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in…

Can something divide one of its divisors?

Posted by Jack M, at math.stackexchange.com, 26 Apr 2014

Let $x$ be an element of a ring and $d$ a divisor of $x$. Can we have $x \mid d$? There's the trivial case where both $x$ and $d…

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