Krdytkyu

This lemma is from Hungerford's Algebra (p.388, Lemma VIII.4.3).

Lemma VIII.4.3
Let $P$ be a prime ideal in a commutative ring $R$ with identity.
If $C$ is a $P$-primary submodule of the Noetherian $R$-module $A$,
then there exists a positive integer $m$ such that $P^m Asubseteq C$.

Proof:
Let $I$ be the annihilator of $A$ in $R$ and consider the ring $overline{R}=R/I$.
Denote the coset $r+Iin overline{R}$ by $overline{r}$.
Clearly $Isubseteq {rin Rmid rAsubseteq C}subseteq P$,
whence $overline{P}=P/I$ is an ideal of $overline{R}$.
$A$ and $C$ are each $overline{R}$-modules with $overline{r}a=ra$ ($rin R, ain A$).
We claim that $C$ is a primary $overline{R}$-submodule of $A$.

If $overline{r}ain C$ with $rin R$ and $ain A-C$,
then $rain C$.
Since $C$ is a primary $R$-submodule,
$r^n Asubseteq C$ for some $n$,
whence $overline{r}^n Asubseteq C$ and $C$ is $overline{R}$-primary.
Since ${overline{r}in overline{R}mid overline{r}^k Asubseteq Ctext{ for some }k>0}={overline{r}in overline{R}mid r^k Asubseteq C}={overline{r}in overline{R}mid rin P}=overline{P}$,
$overline{P}$ is a prime ideal of $overline{R}$ and $C$ is a $overline{P}$-primary $overline{R}$-submodule of $A$. (see Theorems VIII.2.9 and VIII.3.2).

Since $overline{R}$ is Noetherian by Lemma VIII.4.2,
$overline{P}$ is finitely generated by Theorem VIII.1.9.
Let $overline{p}_1, ..., overline{p}_s$ ($p_iin P$) be the generators of $overline{P}$.
For each $i$ there exists $n_i$ such that $overline{p}_i^{n_i}Asubseteq C$.
If $m=n_1+cdots+n_s$,
then it follows from Theorems III.1.2(v) and III.2.5(vi) that $overline{P}^m Asubseteq C$.
The facts that $overline{P}=P/I$ and $IA=0$ now impl that $P^m Asubseteq C$.

My Questions

1. I guess the assertion "We claim that $C$ is a primary $overline{R}$-submodule of $A$" was used to prove that "For each $i$ there exists $n_i$ such that $overline{p}_i^{n_i}Asubseteq C$".
   But I think $overline{p}_i^{n_i}Asubseteq C$ can be obtained from $p_iin P$ directly.
   
   My explaination: Since $p_iin P$ and $C$ is a $P$-primary submodule of $A$,
   we have $p_i^{n_i} Asubseteq C$ for some $n_i$.
   It follows that $overline{p}_i^{n_i} A=p_i^{n_i} Asubseteq C$.
   
   That is, I don't understand why the author spent a bunch of times to prove the unnecessary assertion.




2. Where is the condition "$IA=0$" used?




3. There is a typo in the sentence "whence $overline{r}^n Asubseteq C$ and $C$ is $overline{R}$-primary". It should be "$overline{P}$-primary"

This lemma was used to proved the Krull Intersection Theorem.

The proof of Krull Intersection Theorem.

• Milne's A Primer of Commutative Algebra. Theorem 3.16 on page 13.


• Hungerford's Algebra. Theorem 4.4 on page 389.

The proof of Krull Intersection Theorem by using the Artin-Rees Lemma.

• Atiyah's Introduction to Commutative Algebra. Theorem 10.17 on page 110.


• Eisenbud's Commutative Algebra. Corollary 5.4 on page 150.


• Bosch's Algebraic Geometry and Commutative Algebra. Theorem 2 on page 71.


• Matsumura's Commutative Ring Theory. Theorem 8.9 on page 60.


• 
  The Stack Project. Lemma 10.50.4. https://stacks.math.columbia.edu/tag/00IP
  

The proof of Krull Intersection Theorem by using the Primary Decomposition Theorem.

• Sharp's Steps in Commutative Algebra. Corollary 8.25 on page 159.


• Milne's A Primer of Commutative Algebra. Remark 19.14 on page 92.