We know that if M is an R-module, A,B,C are submodules of M and C is subset of A then $ A \cap (B+C) = (A \cap B) +C $ What if we use $ \oplus $ instead of +. Is it true to write that equation again? Or should we add more things to make it true for direct sum? Thank you for any help.
2026-03-31 05:21:36.1774934496
Modular law for direct summands
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No, consider $M=\mathbb{Z}$ as a $\mathbb{Z}$-module. Take \begin{align} A=6\mathbb{Z},&& B=3\mathbb{Z},&&C=12\mathbb{Z}. \end{align} Then $C\subseteq A$, but $A\cap B=6\mathbb{Z}$ is not a direct summand of $C=12\mathbb{Z}$.