41:45 It may confuse to took $x\in M_p$ as $x_1+...+x_r$ (Since x_1, ...,x_r are preserve for a basis of M). The processor of the proof is assume that $x = y_1+...+y_r$ with $y_i in M_{p_i}$ and $(p^k)x = 0$. then we can show that $y_i = 0 \forall i$ then $ x = 0$ i.e. $M_p$ is a zero module.
At 41:45 you took x (in M_p) as x_1+...+x_r with x_i in M_{p_i}. This notation becomes a source of confusion resulting in an error in the final proof that M_p=(0) for almost all primes p. At 43:45 you have proved that p^k is in Ann(x_i) and perhaps mistook this x_i as the generator x_i of M which is not the same thing (writing x=y_1+...+y_r would be more apt) and continued by taking Ann(x_i)=(d_i). The correct way proceeds as follows: So x=y_+...+y_r with y_i in M_{p_i} shows p^k is in Ann(y_i) and also y_i being an element of M_{p_i} is killed by some (p_i)^l for some l>0. Hence both (p_i)^l and p^k are in Ann(y_i) and so (1)=(p^k,p_i^l) is contained in Ann(y_i) implying Ann(y_1)=(1), the whole ring. Hence y_i=1.y_i=0...so x=0...
41:45 It may confuse to took $x\in M_p$ as $x_1+...+x_r$ (Since x_1, ...,x_r are preserve for a basis of M). The processor of the proof is assume that $x = y_1+...+y_r$ with $y_i in M_{p_i}$ and $(p^k)x = 0$. then we can show that $y_i = 0 \forall i$ then $ x = 0$ i.e. $M_p$ is a zero module.
At 41:45 you took x (in M_p) as x_1+...+x_r with x_i in M_{p_i}. This notation becomes a source of confusion resulting in an error in the final proof that M_p=(0) for almost all primes p. At 43:45 you have proved that p^k is in Ann(x_i) and perhaps mistook this x_i as the generator x_i of M which is not the same thing (writing x=y_1+...+y_r would be more apt) and continued by taking Ann(x_i)=(d_i). The correct way proceeds as follows: So x=y_+...+y_r with y_i in M_{p_i} shows p^k is in Ann(y_i) and also y_i being an element of M_{p_i} is killed by some (p_i)^l for some l>0. Hence both (p_i)^l and p^k are in Ann(y_i) and so (1)=(p^k,p_i^l) is contained in Ann(y_i) implying Ann(y_1)=(1), the whole ring. Hence y_i=1.y_i=0...so x=0...