Divisibility detected up to a bounded defect

Let $R$ be a commutative domain which is a discrete valuation ring, and let $\varpi \in R$ be an irreducible element. Let $A$ be a commutative $R$-algebra, and let $M$ be an abelian group carrying compatible $R$- and $A$-module structures, the $R$-action being the one induced through $A$ (a scalar tower), such that $M$ is finitely generated over $R$ and has no zero smul-divisors over $R$, i.e. $r \cdot x = 0$ with $r \in R$, $x \in M$ forces $r = 0$ or $x = 0$. Assume the action of $A$ on $M$ is faithful in the sense that any $t \in A$ with $t \cdot x = 0$ for all $x \in M$ is zero. The conclusion asserts the existence of a single natural number $b$, depending only on these data, such that for every natural number $m$ and every $t \in A$: if for each $x \in M$ there is some $y \in M$ with $t \cdot x = \varpi^{m+b} \cdot y$ — that is, $tM \subseteq \varpi^{m+b} M$ — then there is $t' \in A$ with $t = \varpi^{m} \cdot t'$, the scalar action of $R$ on $A$. Note that the defect $b$ is uniform in $m$ and $t$.

An elementary piece of commutative algebra over a discrete valuation ring: a faithful order $A$ inside $\operatorname{End}_R(M)$ need not be saturated, but the failure is bounded, so divisibility of $tM$ by $\varpi^{m+b}$ forces divisibility of $t$ by $\varpi^{m}$ in $A$. It is used in the Hecke-algebra arguments on the Tate module of a modular Jacobian, being cited by ModularCurve.exists_generator_tateModule_inf_pi_closure_inertia_smul_sub_and_smul_eisensteinTorsionBar_eq_zero and by ModularCurve.exists_latticeRestrict_heckeEvalForms_mem_span_two_pow_of_forall_smul_eq_zero.