Let $R$ be a commutative ring and let $M$ be an infinite torsion $R$-module. Does $M$ have an infinite sequence of finite submodules of increasing cardinality?

If not, is it true given additional conditions on $R$?

Edit: My motivation was the case $R=\mathbb{Z}$, when $M$ is a torsion Abelian group. In this case, I believe the answer is affirmative: simply consider any infinite sequence $\emptyset \ne E_1 \subset E_2 \subset \cdots \subset M$: it induces another infinite sequence $\{0\} \le H_1 \le H_2 \le \cdots \le M$ whose elements $H_i$ are submodules of $M$ generated by the subsets $E_i$ of $M$. Having nested generating sets, $H_1,H_2,\ldots$ have nondecreasing cardinalities $|H_i|$. Additionally, we can see inductively that each $H_i$ has finitely many elements, since every torsion element of an Abelian group has finite order. It follows that this ascending chain of submodules cannot terminate, so we can pick an infinite subsequence $(K_1,K_2,\ldots)$ of $(H_1,H_2,\ldots)$ whose cardinalities $|K_i|$ are increasing.

However, I have been unable to generalize this construction to other rings.


1 Answer 1


If $R=\mathbb C[X]$, and $M=\mathbb C[X]/(X^2)$, then there is no such chain for the simple reason that $M$ is a noetherian $R$-module, so any ascending chain of submodules must stop.


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