Rational cyclicity gives cyclicity after a bounded power of varpi

Let $R$ be a commutative ring that is a domain and 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 module structures over $R$ and over $A$ (the $R$-action factoring through $A$ in the sense of a scalar tower) such that $M$ is finitely generated as an $R$-module. Let $g \in M$, and suppose that $g$ generates $M$ over $A$ up to $R$-torsion: for every $x \in M$ there exist $d \in R$ with $d \neq 0$ and $t \in A$ such that $d \cdot x = t \cdot g$. The conclusion is that a single exponent suffices uniformly: there exists a natural number $a$ such that for every $x \in M$ there is $t \in A$ with $\varpi^{a} \cdot x = t \cdot g$, that is, $\varpi^{a} M \subseteq A g$. No torsion-freeness of $M$, faithfulness of the $A$-action, or finiteness of $A$ over $R$ is assumed.

This is the rank-one half of the elementary lattice lemma used when a module that is rationally cyclic over an order is compared with the submodule generated by one element: rational cyclicity upgrades to cyclicity up to a bounded power of the uniformiser. It is used in the construction of a generator of the relevant part of the Tate module of a modular Jacobian, in ModularCurve.exists_generator_tateModule_inf_pi_closure_inertia_smul_sub_and_smul_eisensteinTorsionBar_eq_zero.