Question 1179185
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Look into these links/sources


https://yutsumura.com/if-r-is-a-noetherian-ring-and-frto-r-is-a-surjective-homomorphism-then-r-is-noetherian/


and find there the proof to the statement you need.




(It is the area  (part of Abstract Algebra)  which I studied during my years at the university many years ago . . . )