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Bookmark or Share- abstract algebra - Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong ...Suppose $\gcd(m, n) = d\gt 1$. Then $\frac{mn}{d}$ is divisible by both $\,m \text { and}\;n.\;$ Therefore, for any $(r, s) \in \mathbb Z_m \times \mathbb Z_n$ we ...
- modules - How to prove that $R/I \otimes_R M \cong M / IM ...Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
- Difference between "≈", "≃", and "≅" - Mathematics Stack Exchange$\cong$ is used to show a congruency between two mathematical expressions, which could be geometrical, topological, and when using modulo arithmetic you can get different numbers that are congruent, e.g., $5 \text{ mod } 3 \cong 11 \text{ mod } 3$ (although this is also written as $\equiv$). In LaTeX it is coded as \cong.
- A nonsplit short exact sequence of abelian groups with $B \\cong A ...$\begingroup$ And on this specific problem, we know that exactness is stalk-local, and I believe that under some finiteness condition (like Noetherian, finitely presented, etc), split-ness is also stalk-local (since we have some base-change property of $\operatorname{Ext}$, maybe one can find some elementary proof, say passage from stalks to a fundamental open set).
- Proof of $(\\mathbb{Z}/m\\mathbb{Z}) \\otimes_\\mathbb{Z} (\\mathbb{Z ...Stack Exchange Network. Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
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