Recall that a (non-unital, non-commutative) ring $R$ is left s-unital if for every $r\in R$, we have $r\in Rr$.

Consider the following conditions:

• There is a nonzero integer $m$ such that for all $r\in R$, $mr\in Rr$.
• For each $r\in R$, there is a nonzero integer $m_r$ such that $m_rr\in Rr$.

For example, $2\mathbf Z$ has the former property, while $\bigoplus_n n\mathbf Z$ has the latter.

Note that first condition is satisfied by all rings of finite characteristic, the second one is satisfied by all $\mathbf Z$-torsion rings.)

I suppose you could naturally generalise it to modules: if $R,S$ are rings and $M$ is simultaneously a left $R$-module and a left $S$-module, then the corresponding properties would be:

• There is a regular/nonzero $s\in S$ such that for all $v\in M$ we have $sr\in Rv$.
• For each $v\in M$ there is a regular/nonzero $s\in S$ such that $sv\in Rv$.

Do these properties have established names?

Recall that a (non-unital, non-commutative) ring $R$ is left s-unital if for every $r\in R$, we have $r\in Rr$.

Consider the following conditions:

• There is a nonzero integer $m$ such that for all $r\in R$, $mr\in Rr$.
• For each $r\in R$, there is a nonzero integer $m_r$ such that $m_rr\in Rr$.

Question: Do these properties have established names? Have they been studied?

Some observations:

• For example, $2\mathbf Z$ has the former property, while $\bigoplus_n n\mathbf Z$ has the latter, but not the former.
• For right Noetherian rings, the two are equivalent.
• The first condition is satisfied by all rings of finite characteristic, the second one is satisfied by all $\mathbf Z$-torsion rings.

I suppose you could naturally generalise it to modules: if $R,S$ are rings and $M$ is simultaneously a left $R$-module and a left $S$-module, then the corresponding properties would be:

• There is a regular/nonzero $s\in S$ such that for all $v\in M$ we have $sv\in Rv$.
• For each $v\in M$ there is a regular/nonzero $s\in S$ such that $sv\in Rv$.