Let R be Noetherian and let B be an R-module. If P is a prime ideal such that P = ann x for some nonzero x ϵ B (see Exercise 7), then P is called an associated prime of B.

(a) If B ≠ 0, then there exists an associated prime of B. 

(b) If B≠ O and B satisfies the ascending chain condition on submodules, then there exist prime ideals P_1, ... , P_r-1 and a sequence of submodules B = B₁ ⊃ B₂ ⊃ · · · ⊃ B_r = 0 such that B_i/ B_i+1 ≅ R/ P_i for each i < r.

Data from exercise 7

Transcribed Image Text:

7. If B is an R-module and x e B, the annihilator of x, denoted ann x, is {re R| rx = 0). Show that ann x is an ideal.