Question

Let I be an ideal in $k[x_1, \dots, x_n]$, and let $f_1, f_2, \dots, f_r \in k[x_1, \dots, x_n]$. Prove that the followings are equivalent: i) $f_1, \dots, f_r \in I$ ii) $<f_1, \dots, f_r> \subset I$

          Let I be an ideal in $k[x_1, \dots, x_n]$, and let $f_1, f_2, \dots, f_r \in k[x_1, \dots, x_n]$. Prove that the followings are equivalent:
i) $f_1, \dots, f_r \in I$
ii) $<f_1, \dots, f_r> \subset I$

Let I be an ideal in k[x1, …, xn], and let f1, f2, …, fr ∈ k[x1, …, xn]. Prove that the followings are equivalent:
i) f1, …, fr ∈ I
ii) <f1, …, fr> ⊂ I

Added by Mariah M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Sri K

04/27/2023

Step 1

i) $\implies$ ii): Assume $f_1, \dots, f_r \in I$. Let $g \in <f_1, \dots, f_r>$. Then $g$ can be written as a linear combination of $f_1, \dots, f_r$ with coefficients in $k[x_1, \dots, x_n]$. That is, $g = h_1 f_1 + \dots + h_r f_r$ for some $h_1, \dots, h_r \in Show more…

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Let $I$ be an ideal in $k[x_1, \dots, x_n]$, and let $f_1, f_2, \dots, f_r \in k[x_1, \dots, x_n]$. Prove that the followings are equivalent: i) $f_1, \dots, f_r \in I$ ii) $<f_1, \dots, f_r> \subset I$