Answer to Question #144113 in Abstract Algebra for anjali g

Question #144113
If R is noetherian and I is a ideal show that R/I is noetherian.
1
Expert's answer
2020-11-16T07:08:52-0500

Theorem. Let "R" be a Noetherian ring. If "I" is an ideal of "R" then "R\/I" is Noetherian.

Proof. Let "J_1 \u2282 J_2 \u2286 \u00b7\u00b7\u00b7 \u2286 J_i \u2286 J_{i+1} \u2286 \u00b7\u00b7\u00b7" be an ascending chain of ideals in "R\/I". By the Correspondence Theorem, "R" has ideals "K_i (i = 1, 2, . . . )"  such that "I \u2286 K_i" and "K_i\/I = J_i" and "K_i \u2286 K_{i+1}" for all "i".  Then "I \u2286 K_1 \u2286 K_2 \u2286 \u00b7\u00b7\u00b7 \u2286 K_i \u2286 K_{i+1} \u2286 \u00b7\u00b7\u00b7" is an ascending chain of ideals of "R". Since "R" is Noetherian, there is some "N" such that "K_i = K_N" whenever "i \\geqslant N," so "J_i = K_i\/I = K_N\/I = J_N" for "i \\geqslant N". Hence "R\/I" is Noetherian.


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