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Feb 10, 2025 · Let $R$ be a ring with unity, and let $e$ be an idempotent element of $R$ such that $e^2 = e$. If $e$ is a central idempotent of $R$, then we obtain the following ring …
Originally you asked for $\mathbb {Z}/ (m) \otimes \mathbb {Z}/ (n) \cong \mathbb {Z}/\text {gcd} (m,n)$, so any old isomorphism would do, but your proof above actually shows that $\mathbb …
Sep 28, 2024 · Let $G$ be a group with some completely reducible finite-dimensional representations $V,W$, and let $\\mathbf{1}$ be the trivial irreducible representation. I'm ...
This approach uses the chinese remainder lemma and it illustrates the 'unique factorization of ideals' into products of powers of maximal ideals in Dedekind domains: It follows $-1 \cong 10 …
Jan 1, 2025 · I went through several pages on the web, each of which asserts that $\operatorname {Aut} A_n \cong \operatorname {Aut} S_n \; (n\geq 4)$ or an equivalent …
In mathematical notation, what are the usage differences between the various approximately-equal signs '≈', '≃', and '≅'? The Unicode standard lists all of them inside the Mathematical …
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Aug 9, 2025 · Are you sure that $\left\ {\pmatrix { a & b\\0&a^2}\mid a,b\in K, a\ne 0\right\}\cong (K,+)$?
Nov 13, 2015 · A symbol I have in my math homework looks like a ~ above a =. (That is, $\\cong$.) What does this mean? I'm studying Congruency at the moment if that helps.
In geometry, $\cong$ means congruence of figures, which means the figures have the same shape and size. (In advanced geometry, it means one is the image of the other under a …
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