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fixed latex rendering (racther hackily), spacing, and wording (slightly)
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Ramsey
Ramsey
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Let $(n_1,\ldots, n_i,\ldots) $$(n_1,\ldots, n_i,\ldots)$ be an infinite tuple of nonnegative integers.Is Is there an abstract number ring $D$ of a given characteristic $p>0$ and $I_1,\dots, I_n , \ldots$ its nonzero ideals  ( Byby assumption $D/I_i$ are finite) such that #$D/I_i=n_i$ for each $i$ $\# D/I_i=n_i$?

Let $(n_1,\ldots, n_i,\ldots) $ be an infinite tuple of nonnegative integers.Is there an abstract number ring $D$ of a given characteristic $p>0$ and $I_1,\dots, I_n , \ldots$ its nonzero ideals( By assumption $D/I_i$ are finite) such that for each $i$ $\# D/I_i=n_i$?

Let $(n_1,\ldots, n_i,\ldots)$ be an infinite tuple of nonnegative integers. Is there an abstract number ring $D$ of a given characteristic $p>0$ and $I_1,\dots, I_n , \ldots$ its nonzero ideals  (by assumption $D/I_i$ are finite) such that #$D/I_i=n_i$ for each $i$?

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user16974
user16974

Let $D$$(n_1,\ldots, n_i,\ldots) $ be anyan infinite tuple of nonnegative integers.Is there an abstract number ring $D$ of a given characteristic $0$ or $p>0$. Let and $I_1,\dots, I_n , \ldots$ be its nonzero ideals.( By assumption $D/I_i$ are finite. Let $(n_1,\ldots, n_i,\ldots) $ be an infinite tuple of nonnegative integers. Is it possible) such that for each $i$ $\# D/I_i=n_i$?

Let $D$ be any abstract number ring of a given characteristic $0$ or $p>0$. Let $I_1,\dots, I_n , \ldots$ be its nonzero ideals. By assumption $D/I_i$ are finite. Let $(n_1,\ldots, n_i,\ldots) $ be an infinite tuple of nonnegative integers. Is it possible that for each $i$ $\# D/I_i=n_i$?

Let $(n_1,\ldots, n_i,\ldots) $ be an infinite tuple of nonnegative integers.Is there an abstract number ring $D$ of a given characteristic $p>0$ and $I_1,\dots, I_n , \ldots$ its nonzero ideals( By assumption $D/I_i$ are finite) such that for each $i$ $\# D/I_i=n_i$?

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user16974
user16974

Abstract number ring of any characteristic

Let $D$ be any abstract number ring of a given characteristic $0$ or $p>0$. Let $I_1,\dots, I_n , \ldots$ be its nonzero ideals. By assumption $D/I_i$ are finite. Let $(n_1,\ldots, n_i,\ldots) $ be an infinite tuple of nonnegative integers. Is it possible that for each $i$ $\# D/I_i=n_i$?