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If R is a ring and let S be it's subring . If R satisfies Ascending Chain condition,then does S also satisfies Ascending Chain condition?

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No. For example, take $\mathbb{C}[x_1,x_2,\dots]$ as a subring of its field of fractions $\operatorname{Frac}(\mathbb{C}[x_1,x_2,\dots])$. The latter is a field, and hence satisfies the ascending chain condition. The former does not satisfy the ascending chain condition, since $(x_1)\subset (x_1,x_2)\subset (x_1,x_2,x_3)\subset\cdots$ never stabilizes.

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  • $\begingroup$ @ArturoMagidin Yes, thank you for catching this. I edited it accordingly. $\endgroup$ Commented May 22 at 2:23

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