$^\ast \mathbb R$: the field of hyperreal numbers |
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$C([0,1])$, the ring of continuous real-valued functions on the unit interval |
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$F_2[\mathcal Q_8]$ |
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$F_2[x,y]/(x,y)^2$ |
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$F_p(x)$ |
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$k\langle x,y \rangle/(xy-1)$: the Toeplitz-Jacobson algebra |
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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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$k[x;\sigma]/(x^2)$ (Artinian) |
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$k[x;\sigma]/(x^2)$ (not right Artinian) |
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$k[x,x^{-1};\sigma]$ |
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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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$\mathbb H$: Hamilton's quaternions |
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$\mathbb Q$: the field of rational numbers |
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$\mathbb Q\langle a,b\rangle/(a^2)$ |
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$\mathbb Q[\mathbb Q]$ |
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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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$\mathbb Z$: the ring of integers |
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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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$\mathbb Z\langle x,y\rangle/(y^2, yx)$ |
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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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$\mathbb Z/(p)$, $p$ an odd prime |
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$\mathbb Z/(p^k)$, $p$ a prime, $k>1$ |
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$\mathbb Z_S$, where $S=((2)\cup(3))^c$ |
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$\mathbb Z[\sqrt{-5}]$ |
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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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$M_n(F_q)$ |
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$M_n(k)$ |
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$p$-adic integers: $\mathbb Z_p$ |
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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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$T_2(F_2)$ |
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$T_n(F_q)$ |
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$T_n(k)$: the upper triangular matrix ring over a field |
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$\widehat{\mathbb Z}$: the profinite completion of the integers |
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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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Algebra of differential operators on the line (1st Weyl algebra) |
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Bass's right-not-left perfect ring |
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Berberian's incompressible Baer ring |
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Bergman's example showing that "compressible" is not Morita invariant |
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Bergman's exchange ring that isn't clean |
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Bergman's non-unit-regular subring |
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Bergman's primitive finite uniform dimension ring |
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Bergman's right-not-left primitive ring |
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Bergman's unit-regular ring |
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Chase's left-not-right semihereditary ring |
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Clark's uniserial ring |
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Cohn's non-IBN domain |
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Cohn's right-not-left free ideal 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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Cozzens simple, left principal, right non-Noetherian domain |
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Cozzens' simple V-domain |
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Custom Krull dimension valuation ring |
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Faith-Menal counterexample |
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field of $p$-adic numbers |
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Field of algebraic numbers |
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Field of constructible numbers |
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Finitely cogenerated, not semilocal ring. |
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Full linear ring of a countable dimensional right vector space |
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Goodearl's simple self-injective operator algebra |
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Goodearl's simple self-injective von Neumann regular ring |
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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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Hurwitz quaternions |
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Interval monoid ring |
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Kasch not semilocal ring |
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Kolchin's simple V-domain |
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Leavitt path algebra of an infinite bouquet of circles |
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Left-not-right Noetherian domain |
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Left-not-right pseudo-Frobenius ring |
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Left-not-right valuation domain |
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Lipschitz quaternions |
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Local right-not-left Kasch ring |
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Malcev's nonembeddable domain |
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McGovern's commutative Zorn ring that isn't clean |
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Michler & Villemayor's right-not-left V ring |
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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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Nakayama's quasi-Frobenius ring that isn't Frobenius |
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Nielsen's semicommutative ring that isn't McCoy |
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Noetherian domain that is not N-1 |
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Noetherian ring that is not Grothendieck and not Nagata |
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Non-Artinian simple ring |
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Non-symmetric $2$-primal ring |
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O'Meara's infinite matrix algebra |
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Osofsky's $32$ element ring |
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Page's left-not-right FPF ring |
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Perfect non-Artinian ring |
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Perfect ring that isn't semiprimary |
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Pseudo-Frobenius, not quasi-Frobenius ring |
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Puninski's triangular serial ring |
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reduced $I_0$ ring that is not exchange |
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reduced exchange ring which is not semiregular |
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Reversible non-symmetric ring |
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Right-not-left ACC on annihilators triangular ring |
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Right-not-left Artinian triangular ring |
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Right-not-left coherent ring |
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Right-not-left Kasch ring |
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Right-not-left Noetherian triangular ring |
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Right-not-left nonsingular ring |
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Right-not-left simple injective ring |
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Ring of holomorphic functions on $\mathbb C$ |
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Samuel's UFD having a non-UFD power series ring |
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Semicommutative $R$ such that $R[x]$ is not semicommutative |
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Shepherdson's domain that is not stably finite |
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Simple, connected, Noetherian ring with zero divisors |
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Simple, non-Artinian, von Neumann regular ring |
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Šter's clean ring with non-clean corner rings |
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Šter's counterexample showing "clean" is not Morita invariant |
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Varadarajan's left-not-right coHopfian ring |
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