$A$, $B$, $C$ are finite generated modules over $F$, where $F$ is PID.
$A \oplus B \cong A \oplus C$
Is $B \cong C$?
Please give me some direction, i don`t have any specific solution.
2026-03-31 21:09:10.1774991350
Isomorphism of finite generated modules
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You can use the structure theorem for f.g. modules over a PID. Each has the form $$M\cong F^n\oplus\bigoplus_{i=1}^n F/(p_i^{n_i}F)$$ where the $p_i$ are irreducibles. Uniqueness holds: $M$ determines $n$ and the $p_i$ and $n_i$ up to trivialities such as order and replacing an irreducible by an associate.