Leave $ R $ be a unitary, possibly noncommutative ring and $ s in R $. For a right $ R $module $ M $, define $ Ms = {ms mid m in M } $; this is an additive subgroup of $ M $, which is a module on the centralizer of $ s $. For an ordinal $ alpha $, define $ Ms ^ alpha $ by transfinite induction: $ Ms ^ 0 = M $, $ Ms ^ { alpha + 1} = (Ms ^ alpha) s $and take intersections in the limit ordinals. Then the groups $ Ms ^ alpha $ it must eventually stabilize; say that the $ s $divisibility range of $ M $ is the minimum $ alpha $ such that $ Ms ^ alpha = Ms ^ { alpha + 1} $. Consider the following conditions:

For some $ s in R $, the $ s $divisibility range $ R in Mod_R $ is infinite (i.e. $ R $ it's not strongly $ pi $regular).

For some $ s in R $ and something $ M in Mod_R $, the $ s $divisibility range $ M $ It's infinite.

For some $ s in R $, there is a right $ R $– arbitrarily large modules $ s $divisibility range.
Obviously, $ 3 Rightarrow 2 Leftrightarrow 1 $. I suspect 1,2,3 are equivalent, and I think I can show this when $ R $ it is commutative, through a kind of "generalized Prufer module" construction. But I'm not sure when $ R $ It is not commutative.
Question 1: Make $ 3 Rightarrow 1 $ when $ R $ is it noncommutative?
There is another condition that these seem to be related to. Leave $ m in M $ and $ n in N $ be two elements of law $ R $modules. Say it $ m in M, n in N $ are weakly equivalent if there are maps on the right $ R $modules $ f: M a N $ and $ g: N a M $ such that $ f (m) = n $ and $ g (n) = m $. This is an equivalence relationship in elements of law $ R $modules. Consider the condition:
 There is an adequate class of elements of law $ R $modules, no two of which are weakly equivalent.
Why $ f (Ms ^ alpha) subseteq N s ^ alpha $ for each map on the right $ R $modules $ f: M a N $we have that $ 3 Rightarrow 4 $.
Question 2: Make $ 4 Rightarrow 3 $?
The "compound" of questions 1 and 2, that is, the question of whether $ 4 Rightarrow 1 $, can be expressed in a contradictory way as: if $ R $ is strongly $ pi $regular then it does $ R $ have only one set of weak equivalence classes of elements of $ R $modules?