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Sam Hardin 2022-07-02 Answered
Content ( f g ) Content ( f ) Content ( g ) rad ( Content ( f g ) )
to deduce that if Content ( f ) contains a nonzerodivisor of R, then f is nonzerodivisor of S = R [ x 1 , . . . , x r ].
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Answers (1)

Elias Flores
Answered 2022-07-03 Author has 24 answers
By hypothesis, there is some nonzero h R such that f h = 0. By replacing R with the subring of R generated by the homogeneous parts of f and h, we may assume that R is finitely generated; in particular, R is a Noetherian ring by the Hilbert basis theorem. Since f is a zerodivisor, it is contained in some associated prime P Ass R ( R ). By the above lemma, P is the annihilator of a homogeneous element. The claim follows.
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