Associated elements in groups and modules [closed]

We call elements of a commutative ring that divide each other associated (associates).

We can define associates for algebraic structures other than rings as elements that "generate" each other.

The term "generate" will be defined by the structure itself: for groups that will be "powers of each other", "multiples of each other" for rings, "scales of each other" for modules, etc.

Are there interesting properties of such elements in groups, modules, etc., similar to how they appear in gcd, lcm, irreducible elements in rings?

One of the reasons behind the question is that when we say "there is a generator" of a group, or "a basis element" of a module, there is actually a set of interchangeable associated elements each of which can be taken as a "generator" or "basis element" of the structure.

Does it make sense to say "class of associates" instead of "generator", "basis class" instead of "basis element" for an algebraic structure?

And if "yes", isn't it better to generalize associates as elements that "generate the same subset" of an algebraic structure rather than "generate each other"?

This a well-known notion in semigroup theory, but it has a different name. Let $$M$$ be a (not necessarily commutative) monoid. Two elements $$x$$ and $$y$$ of $$M$$ are said to be $$\mathcal{J}$$-equivalent if there exist $$a,b,c,d \in M$$ such that $$x = ayb$$ and $$y =cxd$$. The $$\mathcal{J}$$-relation is one of the five Green's relations.
• It looks like it is not the same: all elements of a group are $\mathcal{J}$-equivalent, but not equivalent as group generators (not associates in my terms). Am I missing something? Thank you. Sep 21, 2019 at 23:08
• I just used the definition given in the first line of your question. Note that two elements are $\mathcal{J}$-equivalent if they divide each other. Sep 22, 2019 at 3:00
• The reason this conversation ground out is that you want "generates" to have a clear meaning dictated by the algebraic structure, but it doesn't. Even inside a given algebraic structure there can be more than one meaning for "generates". For example, in a ring $R$, we can ask if $a,b\in R$ generate the same ideal (usual meaning of "associates") or if they generate the same subring. In a group, you can ask if $a$ and $b$ generate the same subgroup or the same sub-semigroup. Feb 8, 2020 at 14:17