## Finitely generated modules

Let $f:A\to B$. Let $A\subseteq B$ be commutative rings with $A$ a subring of $B$.

• A finitely generated $A$-module is an $A$-module $M$ such that there exists $n\in {ℤ}_{>0}$ and a surjective $A$-module homomorphism such that $M$ is isomorphic to a quotient of ${A}^{\oplus n}\to M$.
• A finitely generated $A$-algebra is an $A$-algebra $C$ such that there exists $n\in {ℤ}_{>0}$ and a surjective homomorphism of commutative $A$-algebras $A\left[{x}_{1},\dots {x}_{n}\right]\to C$.
• The ring $B$ is a finite $A-$algebra if $B$ is a finitely generated $A-$module.
• The ring $B$ is an $A-$algebra of finite type if $B$ is a finitely generated $A-$algebra.
• The map $f$ is finite if $f\left(A\right)$ is a finitely generated $A-$module.
• The map $f$ is of finite type if $f\left(A\right)$ is a finitely generated $A-$algebra.

HW: Show that an $A$-module $M$ is finitely generated if and only if there is a finite subset $S\subseteq M$ such that $M=R\text{-span}\left(S\right)$.

Let $A\subseteq B$ be commutative rings with $A$ a subring of $B$.

• An element $b\in B$ is integral over $A$ if $f\left(b\right)=0$ for some monic polynomial $f\left(x\right)\in A\left[x\right]$.
• The ring $B$ is integral over $A$, or $B$ is an integral extension of $A$, if every element of $B$ is integral over $A$.
• The integral closure of $B$ in $A$ is the largest subring of $B$ which is integral over $A$.
• The ring $A$ is integrally closed in $B$ if $A=\left\{b\in B\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}b\phantom{\rule{.5em}{0ex}}\text{is integral over}\phantom{\rule{.5em}{0ex}}A\right\}$.
• An integral domain $R$ is integrally closed if $R$ is integrally closed in its field of fractions.
• An algebraic integer is an element $\alpha \in ℂ$ which is integral over $ℤ$.

HW:Show that if $A\subseteq B$, and $N$ is finitely generated as a $B-$module and $B$ is finitely generated as an $A-$module then $N$ is finitely generated as an $A-$module.

HW:Show that if $A\subseteq B$ and $b\in B$ then $b$ is integral over $A$ if and only if $A[b]= im(evb: A[x]→B)$ is finitely generated as an $A-$module.

HW:Show that if $A\subseteq B$ and ${b}_{1},\dots ,{b}_{k}\in B$ are integral over $A$ then $A\left[{b}_{1},\dots ,{b}_{k}\right]$ is a finitely generated $A$-module.

HW:Show that if $A\subseteq B$ then the integral closure of $A$ in $B$ is $C=\left\{b\in B\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}b\phantom{\rule{.5em}{0ex}}\text{is integral over}\phantom{\rule{.5em}{0ex}}A\right\}$.

HW:Show that if $A\subseteq B$ then the integral closure $C$ of $A$ in $B$ is the largest subring of $B$, $A\subseteq C\subseteq B$, which is finitely generated as an $A-$module.

HW: WE NEED A BETTER NOTATION FOR THE INTEGRAL CLOSURE OF A IN B; WHAT DOES BOURBAKI USE??

Example. $ℤ$ is integrally closed.

Let $A\subseteq B$ be an integral extension.

1. Let $𝔭$ be a prime ideal in $A$ and let ${B}_{𝔭}={A}_{𝔭}{\otimes }_{A}B.$ Then ${A}_{𝔭}\subseteq {B}_{𝔭}$ is an integral extension.
2. Let $𝔟$ be a prime ideal of $B$ and let $𝔞=𝔟\cap A.$ Then $A/𝔞\subseteq B/𝔟$ is an integral extension.

## Structure of the lattice of submodules

Let $A$ be a ring and let $M$ be an $A-$module.

• The module $M$ is finitely generated if there exists $n\in {ℤ}_{>0}$ and ${m}_{1},\dots ,{m}_{n}\in M$ such that $M= span{m1, …,mn},$ where $\text{span}\left\{{m}_{1},\dots ,{m}_{n}\right\}$ is the $A$-submodule of $M$ generated by $\left\{{m}_{1},\dots ,{m}_{n}\right\}$.
• The module $M$ is simple if it has no submodules except $M$ and $0$.
• A finite composition series of $M$ is a finite chain of submodules $0= M0⊆M1 ⊆⋯⊆Mn =Mwith Mi/Mi-1 simple.$
• The module $M$ is Noetherian if every ascending chain of submodules is eventually constant.
• The module $M$ is Artinian if every descending chain of submodules is eventually constant.
• The ring $A$ is Noetherian if $A$ is Noetherian as an $A$-module.
• The ring $A$ is Artinian if $A$ is Artinian as an $A$-module.

Let $A$ be a ring, let $M$ be an $R-$module and let $N$ be a submodule of $M$. Then

1. If $M$ is finitely generated then $M/N$ is finitely generated.
2. $M$ has a finite composition series if and only if $N$ and $M/N$ have finite composition series.
3. $M$ is Noetherian if and only if $N$ and $M/N$ are Noetherian.
4. $M$ is Artinian if and only if $N$ and $M/N$ are Artinian.

Examples.

1. Let $𝔽$ be a field. An $𝔽-$module is a vector space $V$ over $𝔽$. Any of the conditions (i) $V$ is Noetherian, (ii) $V$ is Artinian, or (iii) $V$ has a finite composition series, are equivalent to $V$ being finite dimensional. An infinite dimensional vector space is neither Noetherian or Artinian.
2. The ring $ℤ$ is Noetherian, but not Artinian.

Let $A$ be a ring and let $M$ be an $A-$module.

1. $M$ has a finite composition series if and only if $M$ is Noetherian and Artinian.
2. $M$ is Noetherian if and only if every submodule of $M$ is finitely generated.
3. If $R$ is Noetherian and $M$ is finitely generated then $M$ is Noetherian.

 Proof. (a) Follows from Theorem 2.4 below. (b) ⇐: Assume that every submodule of $M$ is finitely generated. Let ${N}_{1}\subseteq {N}_{2}\subseteq \cdots$ be an ascending chain. Then $\bigcup {N}_{i}$ is a finitely generated submodule of $M.$ Let ${x}_{1},...,{x}_{k}$ be generators and let ${l}_{1},...,{l}_{k}$ be such that ${x}_{i}\in {N}_{{l}_{i}}.$ Then ${x}_{1},...,{x}_{k}\in {N}_{r}$ where $r=\mathrm{max}\left\{{l}_{1},...,{l}_{k}\right\}.$ So $\bigcup {N}_{i}={N}_{r}$ and ${N}_{r}={N}_{r+1}={N}_{l}$ for all $l>r.$ So $M$ in noetherian. (b) ⇒: Assume that $M$ is noetherian and let $N$ be a submodule of $M.$ Then ${ P⊆N | P is finitely generated }$ has a maximal element ${P}_{\mathrm{max}}.$ If ${P}_{\mathrm{max}}\ne N$ let $x\in N\setminus {P}_{\mathrm{max}}.$ Then $P\subseteq ⟨{P}_{\mathrm{max}},x⟩\subseteq N$ and $⟨{P}_{\mathrm{max}},x⟩$ is finitely generated, which is a contradiction to the maximality of ${P}_{\mathrm{max}}.$ So ${P}_{\mathrm{max}}=N.$ So every submodule of $M$ is finitely generated. $\square$

(Jordan-Hölder theorem.) Let $M$ be an $A-$module.

1. Any two series $0⊆M1⊆ ⋯⊆Mr = Mand 0⊆M1' ⊆⋯⊆ Ms'=M,$ can be refined to have the same length and the same composition factors.
2. $M$ has a composition series if and only if any series can be refined to a composition series.
3. $M$ has a composition series if and only if $M$ is Noetherian and Artinian.
4. If $M$ has a composition series then any two composition series of $M$ have the same length.

 Proof. Suppose $0=M0⊆ M1⊆⋯⊆ Mr=M and 0=M0'⊆ M1' ⊆⋯⊆ Ms'=M,$ are chains of submodules of $M$. Change ${M}_{i}\subseteq {M}_{i+1}$ to $Mi= (M0'+ Mi) ∩Mi+1 ⊆(M1'+ Mi)∩ Mi+1 ⊆⋯⊆ (Ms'+ Mi)∩ Mi+1 =Mi+1$ and change ${M}_{j}^{\text{'}}\subseteq {M}_{j+1}^{\text{'}}$ to $Mj= (M0+ Mj')∩ Mj+1' ⊆(M1+ Mj')∩ Mj+1' ⊆⋯⊆ (Mr+ Mj')∩ Mj+1' =Mj+1'.$ Claim: $(Mj' +Mi-1) ∩Mi (Mj-1' +Mi-1) ∩Mi ≅ (Mi+ Mj-1') ∩Mj' (Mi-1 +Mj-1') ∩Mj' .$ This claim will be established by Lemma 2.6. $\square$

(Modular Law) If $A$, $B$, $C$ are submodules of $M$, and $B\subseteq C$, then $C+(A∩B) =(C+A)∩B.$

 Proof. If $c+a\in C+\left(A\cap B\right)$ then $c+a\in \left(C+A\right)\cap B$. If $b=c+a\in \left(C+A\right)\cap B$ then $b=c+a=c+\left(b-c\right)\in C+\left(A\cap B\right)$. $\square$

(Zassenhaus Isomorphism) If $V\subseteq U$ and $V\text{'}\subseteq U\text{'}$ are submodules of $M$ then $(U+V')∩U' (V+V')∩U' ≅ U∩U' (U∩V')+ (U'∩V) ≅ (U'+V)∩U (V'∩V)∩U .$

(Hilbert basis theorem) Let $R$ be a commutative Noetherian ring. Then $R\left[x\right]$ is a commutative Noetherian ring.

 Proof. To show: Every ideal of $R\left[x\right]$ is finitely generated. Let $𝔞$ be an ideal of $R\left[x\right]$. Let $𝔟 ={ak∈R | f(x)= akxk +⋯+a0∈𝔞 }$ an ideal of $R$. Since $R$ is Noetherian, $𝔟$ is finitely generated. Let ${b}_{1},\dots ,{b}_{n}$ be generators of $𝔟$. Let $f1(x) =b1 xk1 +⋯+b10 ∈𝔞 f2(x) =b2 xk2 +⋯+b20 ∈𝔞, ⋮ fn(x) =bn xkn +⋯+bn0 ∈𝔞,$ be polynomials in $𝔞$ corresponding to the generators of $𝔟$. Then $⟨f1,,… ,fn⟩⊆𝔞$ Let $f\in 𝔞$, $f=a{x}^{k}+\cdots +{a}_{0}$. If $k\ge \mathrm{max}\left({k}_{1},\dots ,{k}_{n}\right)$ then $a= ∑i=1n ribi for some ri∈R, and f- ∑i=1n rifi xk-ri ∈𝔞 and has lower degree.$ Thus, if $f\in 𝔞$ then $f=g+h with g∈𝔞 and deg(h) If $M=R-span{1, x,x2,…, xk-1}, then 𝔞=(𝔞∩M) +⟨f1, …,fn⟩.$ Since $M$ is finitely generated, $M$ is Noetherian. So $𝔞\cap M$ is finitely generated as an $R$-module. Let ${g}_{1},\dots ,{g}_{m}$ generate $𝔞\cap M$. Then ${g}_{1},\dots ,{g}_{m},{f}_{1},\dots ,{f}_{n}$ generate $𝔞$. So $R\left[x\right]$ is Noetherian. $\square$

(Finite generation of invariants.) Let $𝔽$ be a field and let $A$ be a finitely generated $𝔽-$algebra. Let $G$ be a finite group acting on $A$ by automorphisms. Then

1. ${A}^{G}$ is a finitely generated $𝔽-$algebra.
2. $A$ is a finitely generated ${A}^{G}-$module.

 Proof. Let ${a}_{1},\dots ,{a}_{n}$ be generators of $A$ as an $𝔽-$algebra. Let $B= ⟨ c i j ∈AG | c i j are coefficients of ∏g∈G (x-gai) ⟩ .$ Then $B$ is a finitely generated $𝔽-$algebra. So $B$ is a quotient of $𝔽\left[{x}_{1},\dots ,{x}_{m}\right]$ for some $m$. Thus, by the Hilbert basis theorem, $B$ is Noetherian. If $a\in A$ then $a$ satisfies the polynomial $∏g∈G (x-ga) ∈AG[x]$ and so $A$ is an integral extension of ${A}^{G}$. Thus, since ${a}_{1},\dots ,{a}_{n}$ are generators of $A$, we have that $A$ is a finitely generated $B-$module. So ${A}^{G}$ is a finitely generated $B-$module. So ${A}^{G}$ is a finitely generated $𝔽-$algebra. $\square$

## Notes and References

Composition series and the Jordan-Hölder theorem are treated in [Bou, Alg. Ch. I § 4.7] and in [AM, Proposition 6.7]. Chain conditions, Noetherian rings and Artinian rings are covered in [AM] Chapters 6, 7 and 8. The Hilbert basis theorem is [AM, Theorem 7.5] and [Bou, Comm. Alg. Ch. III § 2 No. 10] and the Finite generation of invariants theorem is [AM, Ch. 7 Ex. 5] and [Bou, Comm. Alg. Ch. V § 1 No. 9, Theorem 2]. An alternative, efficient treatment is found in [Ben], where the Hilbert basis theorem is [Ben, Theorem 1.2.4] and the Finite generation of invariants theorem is [Ben, Theorem 1.3.1].

The basics of noetherian and artinian modules and rings are treated in [Bou, Alg. Ch. 8 § 1].

## References

[AM] M. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp. MR0242802.

[Ben] D.J. Benson, Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series 190, Cambridge University Press, Cambridge, 1993. x+118 pp. ISBN: 0-521-45886-2 MR1249931.

[Mac] I.G. Macdonald, Algebraic geometry. Introduction to Schemes, W. A. Benjamin, Inc., New York-Amsterdam 1968 vii+113 pp. MR0238845.