# Tagged Questions

For questions about modules over rings, concerning either their properties in general or regarding specific cases.

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### Characterization of faithfully flat modules

This is an exercise from Rotman, introduction to homological algebra. A right $R$-module $B$ is called faithfully flat if : 1) $B$ is flat 2) If $X$ is a left $R$-module and $B \otimes_R X =0$ ...
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### Help proving a short exact sequence

Show the following sequence is an exact sequence of $\mathbb Z$-modules when $n$ is a positive integer such that $n=rs$: $$0 \to r\mathbb{Z}_n \to \mathbb{Z}_n \to s\mathbb{Z}_n \to 0.$$ should ...
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### Can one prove the existence of tensor product without explicitly constructing it? [duplicate]

R is a ring with 1. We construct tensor product $M \otimes N$ of right R-module $M$ and left R-module $N$ to basically be able to state its universal property that any R-bilinear map from $M\times N$...
Let $R$ be a ring, $A$ an $R$-module, $y$ belong in $R$ such that $1+y$ annihilates $A$. Then for any ideal $I$ containing $y$, prove that $IA=A$.
Let $0 → A → B → C → 0$ be a short exact sequence of $R$-modules. Prove that for any $R$-module $M$ , there is a short exact sequence $0 → A \oplus M → B \oplus M → C → 0$. Can anyone please help me ...