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Akizuki's counterexample |
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Algebraic closure of $F_2$ |
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Algebraic integers |
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$^\ast \mathbb R$: the field of hyperreal numbers |
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Clark's uniserial ring |
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Cohn's Schreier domain that isn't GCD |
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Countably infinite boolean ring |
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Custom Krull dimension valuation ring |
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$F_2[x,y]/(x,y)^2$ |
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Field of algebraic numbers |
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Field of constructible numbers |
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field of $p$-adic numbers |
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Finitely cogenerated, not semilocal ring. |
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$F_p(x)$ |
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Grams' atomic domain which doesn't satisfy ACCP |
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Henselization of $\Bbb Z_{(p)}$ |
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Hochster's connected, nondomain, locally-domain ring |
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Interval monoid ring |
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Kasch not semilocal ring |
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$k[[x]]$ |
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$k[x]$ |
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$k[x^{1/2},x^{1/4},x^{1/8},...]/(x)$ |
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$k[x_1, x_2,\ldots, x_n]$ |
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$k[[x^2,x^3]]$ |
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$k[x,y]/(x^2, xy)$ |
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$k[x,y]/(x^2-y^3)$ |
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$k[x,y]_{(x,y)}/(x^2-y^3)$ |
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$k[x,y,z]/(xz,yz)$ |
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$\mathbb C$: the field of complex numbers |
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$\mathbb F_2[x_1, x_2, x_3\ldots ]/(\{x_i^3\mid i\in \mathbb N\}\cup\{x_ix_j|i\neq j\}\cup\{x_i^2-m\mid i\in \mathbb N\})$ |
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$\mathbb Q[\mathbb Q]$ |
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$\mathbb Q$: the field of rational numbers |
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$\mathbb Q(x)$: rational functions over the rational numbers |
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$\mathbb Q[X,Y]_{(X,Y)}$ |
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$\mathbb R$: the field of real numbers |
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$\mathbb R[x_1, x_2,x_3,\ldots]$ |
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$\mathbb R[x]/(x^2)$ |
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$\mathbb R[x,y]$ completed $I$-adically with $I=(x^2+y^2-1)$ |
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$\mathbb R[x,y]/(x^2+y^2-1)$: ring of trigonometric functions |
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$\mathbb R[x,y,z]/(x^2,y^2, xz,yz,z^2-xy)$ |
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$\mathbb Z/(2)$ |
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$\mathbb Z[\frac{1+\sqrt{-19}}{2}]$ |
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$\mathbb Z[i]$: the Gaussian integers |
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$\mathbb Z/(n)$, $n$ divisible by two primes and a square |
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$\mathbb Z/(n)$, $n$ squarefree and not prime. |
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$\mathbb Z_{(p)}$ |
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$\mathbb Z/(p^k)$, $p$ a prime, $k>1$ |
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$\mathbb Z/(p)$, $p$ an odd prime |
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$\mathbb Z[\sqrt{-5}]$ |
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$\mathbb Z_S$, where $S=((2)\cup(3))^c$ |
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$\mathbb Z$: the ring of integers |
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$\mathbb Z[x]$ |
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$\mathbb Z+x\mathbb Q[x]$ |
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$\mathbb Z[x]/(x^2-1)$ |
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McGovern's commutative Zorn ring that isn't clean |
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Mori but not Krull domain |
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Nagata's Noetherian infinite Krull dimension ring |
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Nagata's normal ring that is not analytically normal |
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Noetherian ring that is not Grothendieck and not Nagata |
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$p$-adic integers: $\mathbb Z_p$ |
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Perfect non-Artinian ring |
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Perfect ring that isn't semiprimary |
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$\prod_{i=1}^\infty F_2$ |
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$\prod_{i=1}^\infty \mathbb Q[[X,Y]]$ |
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Pseudo-Frobenius, not quasi-Frobenius ring |
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reduced exchange ring which is not semiregular |
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reduced $I_0$ ring that is not exchange |
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Ring of holomorphic functions on $\mathbb C$ |
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$\widehat{\mathbb Z}$: the profinite completion of the integers |
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