Let $R$ be a commutative ring with identity such that $R^{n}\simeq R^{n}\bigoplus M$ ($R$-module isomorphism), $n$ is fixed, and $n$ is a natural number. Then $M=\lbrace 0 \rbrace$ ?
2026-03-29 10:12:17.1774779137
Isomorphic modules
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Hint: consider the localization at the prime ideal $P$; then $$ R_P^n\cong R_P^n\oplus M^{\vphantom{}}_P $$ which means $M_P=\{0\}$. (Prove it.)
Since this is for all prime ideals, it follows that…