Understanding Theorem 7.3 Adkins' Algebra I can't understand some of arguments in the following proof which is from the Adkins' Algebra textbook : 

My questions are : 
1- I can't prove Eqs. 7.10 and 7.11, i.e. How Eqs. 7.6 and 7.7 imply Eqs. 7.10 and 7.11?
2- Why if not $Ann(z_i) \subset <p>$ then $R=Ann(z_i) + <p>$? 
3- How we conclude that $k-s = k-t$ (in the third line from ending, reversed)?
PS - I think there are typos in Eqs. 7.12 and 7.13; e.g. $pRw_k \cong$ must be replaced by $pRw_s \oplus$.
 A: for $1$, note that if $R=A\oplus B$ and $I_1$, $I_2$ are ideals of $A$ and $B$ respectively, $R/(I_1 \oplus I_2)=A/I_1 \oplus B/I_2$. To show the isomorphism, consider $u\in R$ which has the unique decomposition $u=u_1+u_2$ in which $u_1\in A$ and $u_2\in B$. Then the isomorphism is constructed as $f(\bar u)=\bar {u_1}+\bar {u_2}$. 
for $2$, $p$ is prime. Suppose $Ann(z_i)=<a_i>$ since R is a PID. Now, $<p>\not\supset<a_i>$ means $p\not| a_i$. As R is a PID $<p>$ is maximal. However, $a_i\not\in <p>$. This implies $<p>+<a_i>\supsetneq<p>$, but the only ideal that strictly contains $<p>$ is , so $<p>+<a_i>=R$.
for $3$, the point is, when you change $M$ to $pM$, $l(pM)=l(M)-1$ so that we can apply the induction hypothesis. As you observed, the number of components annihilated, $s$ and $t$, may be zero. However, $k-s$ and $k-t$, the number of components for $pM$, are equal, according to the induction hypothesis.
One remark regarding the "typo": no there are none. What the text meant was that the prime $p$ annihilates all $Rw_i$ for $i$ from $1$ to $s$. Same for $Rz_i$.
Hope I helped.
