2.1 Basic commutative ring theory

KNOW BASIC DEFINITIONS: (commutative) ring, subring, ring homomorphism, ideal, quotient (or factor) ring, maximal and prime ideal; prime and irreducible elements; (group of) units, associate elements, divisibility, gcd, lcd; field, integral domain, Euclidean domain, PID, UFD, noetherian ring; polynomial ring $R[x]$; rings of fractions.

KNOW THE BASIC LEMMAS: isomorphism theorem(s); $I$ maximal (prime) $\Leftrightarrow R / I$ is a field (integral domain); finite integral domains are fields; Euclidean algorithm (know how to compute the gcd $d$ of $a$ and $b$ as well as coefficients $x$ and $y$ such that $a x+b y=d$ ); universal properties of polynomial rings and rings of fractions.

Know basic CONSTRUCTIONS: direct products of rings; intersections, sums and products of ideals - recall
$I+J:=\{x+y \mid x \in I$ and $y \in J\}$
$I J:=\left\{\sum_{i=1}^{n} x_{i} y_{i} \mid n \in \mathbb{N} ; x_{i} \in I, y_{i} \in J\right.$ for all $\left.1 \leq i \leq n\right\}$
Know examples of Euclidean domains: $\mathbb{Z} ; K[x]$ for a field $K ; \mathbb{Z}[\alpha]$ for $\alpha=i, \sqrt{-2}, \sqrt{2}, \sqrt{3}$.
Know examples of non-PID’s: $\mathbb{Z}[x] ; K[x, y]$ for a field $K ; \mathbb{Z}[\alpha]$ for $\alpha=\sqrt{-3}, \sqrt{-5}$.
Know the "Hierarchy": field $\Rightarrow$ Euclidean domain $\Rightarrow$ PID $\Rightarrow$ UFD $\Rightarrow$ domain
Know some more theorems: Chinese Remainder Theorem; Gauss’ Lemma ( $R$ UFD $\Rightarrow R[x]$ UFD); Hilbert’s Basis Theorem ( $R$ noetherian $\Rightarrow R[x]$ noetherian); in a PID $R$ any nonzero prime ideal is maximal and the $\operatorname{gcd}$ of $a_{1}, \ldots, a_{n}$ is an $R$-linear combination of $a_{1}, \ldots, a_{n}$.

Know Zorn’s Lemma If $S$ is a nonempty partially ordered set such that every chain (linearly ordered subset) $C$ has an upper bound ( $b \in S$ with $b \geq c$ for all $c \in C$ ), then $S$ has at least one maximal element $m(t \in S, t \geq m \Rightarrow t=m)$.
Main application in ring theory: Any proper ideal $I$ of a ring $R$ is contained in a maximal ideal $M$ of $R$. (The important special case $I=\{0\}$ yields the existence of maximal ideals.)

Avoid the following mistakes: If $\phi: R \longrightarrow S$ is a ring homomorphism and $I$ is an ideal in R, $\phi(I)$ needn’t be an ideal in $S$ (it is if $\phi$ is surjective), and if $I$ is prime (or maximal) in $R$, the ideal generated by $\phi(I)$ in $S$ need not be prime (or maximal) in $S$ (even if $\phi$ is surjective); $\mathbb{Z}[x]$ does not admit a Euclidean algorithm (it’s even not a PID), similarly with $K[x, y]$ where $K$ is a field; if in a domain $R$ the gcd of two elements $a$ and $b$ exists and equals 1 , this does not imply that the ideal ( $a, b$ ) equals $R$ (it does if ( $a, b$ ) is principal).

Related Problems

(Unless otherwise stated, all rings $R$ are commutative with unit $1 ; F$ denotes a field.)

1. (April $77 \# 4$ ) (a) If $M$ is a maximal ideal, SHOW $R / M$ is a field. (Done.)
   (b) If $R$ is a Euclidean domain, show every ideal generated by an irreducible element is maximal.

2. (May 78 #5) Is the ring $C[0,1]$ of continuous real-valued functions on the closed interval $[0,1]$ noetherian?

3. (Sep 78 #6) $R$ is called a local ring if it has a unique maximal ideal; a domain is called a valuation domain if for every two elements $a, b$ either $a$ divides $b$ or $b$ divides $a$. (a) Show $R$ is local iff the non-units form an ideal $M$. (b) Show every valuation domain is local. (c) Show a local ring has no idempotents ( $e^{2}=e$ ) other than 1,0 .

4. (Sep 79 #7a; Nov $77 \# 5$ ) Give two examples of non-noetherian rings.

5. (May $80 \# 2$ ) Hilbert’s Theorem says that $F\left[x_{1}, \ldots, x_{n}\right]$ is noetherian. Show that ANY finitely generated commutative $F$-algebra $A$ is noetherian.

6. (Jan $82 \# 6$ ) If $I$ is an ideal of $R$ with $I \cap S=\emptyset$ for some multiplicatively closed subset $S$ of $R$ containing 1, show there exists an ideal $M$ of $R$ containing $I$ and maximal with respect to $M \cap S=\emptyset$. Find an $M$ if $R=\mathbb{Z}, S=\left\{3^{n} \mid n \geq 0\right\}, I=2 \mathbb{Z}$.

7. (Mar 83 #8) Does 7 divide $10^{31}+31^{10}$ ? Why?

8. (Feb $84 \# 3$ ) If $R$ is finite, show every prime ideal is maximal.

9. (1985#3c) If $a, b$ are relatively prime integers, show the ring $\mathbb{Z}_{a b}$ is isomorphic to the direct sum $\mathbb{Z}_{a} \oplus \mathbb{Z}_{b}$ of rings. Show (in $\leq 1$ word) why this implies if $m=p_{1}^{e_{1}} \ldots p_{t}^{e_{t}}$ that $\mathbb{Z}_{m}$ is isomorphic to $\mathbb{Z}_{p_{1}^{e_{1}}} \oplus \cdots \oplus \mathbb{Z}_{p_{t}^{e_{t}}}$.

10. (May 89 #7; Aug 97 #8c) Show an element $a \in R$ belongs to $J=\bigcap$ {maximal ideals of $R\} \Longleftrightarrow 1+r a$ is a unit for all $r \in R$.

11. (May $1990 \# 5$ ) If $R$ is an integral domain with only finite number $n$ of ideals, show $R$ is a field, and give an upper bound for $n$.

12. (May $92 \# 2$ ) If $a, b$ in an integral domain $R$ satisfy $a^{n}=b^{n}, a^{m}=b^{m}$ for $m$ and $n$ relatively prime, show $a=b$. (Do you need that $R$ is an integral domain?)

13. (Jan 92 #6) Prove that an integral domain has the descending chain condition on ideals iff it is a field.

14. (Sept $93 \# 2$ ) If $P$ is a prime ideal which is not maximal, show $P$ has infinitely many cosets in $R$.

15. (Sept $93 \# 5$ ) (a) Let $D$ be a Euclidean domain, with Euclidean function $\delta$ (but do not assume $\delta(a b) \geq \delta(a)$ ). Let $D_{n}=\{a \in D \mid a \neq 0$ and $\delta(a) \geq n\}$. Show that (i) if $b \in D_{0}$ and there exists an $a \in D$ such that $a+D b \subseteq D_{n}$, then $b \in D_{n+1}$; (ii) $\bigcap_{n} D_{n}=\emptyset$.
    (b) Conversely, show that if an integral domain $D$ has a chain of subsets $D \backslash\{0\}=D_{0} \supseteq D_{1} \supseteq D_{2} \supseteq \ldots$ satisfying properties (i) and (ii), then $D$ is Euclidean.

16. Let $R$ be an integral domain, $N: R \backslash\{0\} \longrightarrow\{n>0 \mid n \in \mathbb{Z}\}$ be a function for which (i) $N(1)=1$ and (ii) $N(x y)=N(x) N(y)$ for all $x, y$ in $R \backslash\{0\}$.
    (a) Let $K$ be the field of fractions of $R$. Show that $N$ can be extended in a unique way to a function from $K \backslash\{0\}$ into $\mathbb{Q}$ that still satisfies (i) and (ii) (now for all $x, y \in K \backslash\{0\}$ ).
    (b) Show that $R$ is a Euclidean domain under $N$ iff for each $x \in K \backslash\{0\}$ there is an element $r \in R$ for which $N(x-r)<1$.

17. (Jan 94 #8) Let $Z D$ denote the set of zero divisors of $R$ (including 0 ). Let $\mathcal{I}$ be the set of ideals of $R$ which are contained in $Z D$.
    (a) Show that $R \backslash Z D$ is closed under multiplication
    (b) Show that if $M$ is a maximal member of $\mathcal{I}$, then $M$ is a prime ideal.
    (c) Use Zorn’s Lemma to show that each member of $Z D$ is contained in a maximal member of $\mathcal{I}$.
    (d) Conclude that the set $Z D$ is a union of prime ideals.

18. (Aug $97 \# 8$ ) (a) Show that the set $N$ of all nilpotent elements $z\left(z^{n}=0\right.$ for some $\left.n\right)$ of $R$ forms an ideal (called the nil radical of $R$ ).
    (b) The intersection $J$ of all maximal ideals of $R$ is called the Jacobson radical of $R$; show that $N \subseteq J$.
    (c) Give an example where $N$ and $J$ are different.

19. (Aug $99 \# 4$ ) If $R$ is an integral domain, find all $R$-linear automorphisms of the polynomial ring $R[x]$. (If you can’t do the general case, do the case when $R=F$ is a field.)

20. (Aug $02 \# 5$ ) Let $R$ be a commutative ring, $\mathfrak{a}_{1}, \ldots, \mathfrak{a}_{n}$ be ideals of $R$. If for all $i \neq j$ we have $\mathfrak{a}_{i}+\mathfrak{a}_{j}=R$ then $\mathfrak{a}_{1} \cap \ldots \cap \mathfrak{a}_{n}=\mathfrak{a}_{1} \cdot \ldots \cdot \mathfrak{a}_{n}$. (Hint. Induct, starting from $n=2$.)

21. (Aug 02 #8) Is it true that in the ring $R$ of all continuous real-valued functions on $[0,1]$, any element which is not a zero divisor, is invertible?
    And (May 03 #6): Give an example of a maximal ideal in $R$.
    And (Jan 06 #4): Is this maximal ideal principal?

22. (Aug $05 \# 3$ ) Let $R$ be a commutative ring with 1 (not necessarily a domain) such that any ideal in $R$ is principal. Let $S \subseteq R$ be a multiplicatively closed subset of $R$. Show that any ideal in the ring of fractions $S^{-1} R$ is also principal.

23. (Aug $06 \# 5$ ) Let $R$ be a commutative ring. A radical is an ideal $I \unlhd R$ such that, for any $a \in R$, we have $a \in I$ whenever some power $a^{k} \in I$.
    (a) Show that every prime ideal is radical.
    (b) Assume $I$ is radical and $a \in R$ does not lie in $I$. Show that there exists a prime ideal $P$ that contains $I$ but does not contain $a$. (Hint: use Zorn’s lemma.)

24. (Aug $08 \# 5$ ) Let $R=\mathbb{Z}[i]$ be the ring of Gaussian integers, and let $I=(a, b)$ be the ideal generated by $a=16 i, b=5+3 i$ in $R$.
    a. Systematically find $c \in R$ such that $I=(c)$.
    b. Prove or disprove: $R / I$ is a finite field.

25. (Jan 09 #4) (a) Prove that the ring of Gaussian integers $\mathbb{Z}[i]$ is a Euclidean domain.
    (b) Let $n$ and $m$ be positive integers and assume that $m$ is a product of distinct primes. Prove that the polynomial $f(x)=x^{n}-m$ is irreducible in $\mathbb{Q}[x]$.

26. (Jan $12 \# 3$ ) Let $R=\mathbb{Z}[\sqrt{-2}]$.
    (a) Prove that $R$ is a Euclidean domain. Hint: Use the square of the usual complex norm.
    (b) Write 7 and 11 as products of irreducible elements of $R$. Justify your answer.

27. (Aug $12 \# 4$ ) Let $R$ be a commutative ring with 1 . Let $N$ be the nilradical of $R$, that is, $N$ is the set of all nilpotent elements of $R$ (including 0 ). You may use without proof that $N$ is the intersection of all prime ideals of $R$. Prove that the following conditions are equivalent:
    (i) $R$ has just one prime ideal.
    (ii) $R / N$ is a field.

28. (Jan $13 \# 2$ ) In both parts of this problem $R$ is a commutative domain with 1 and $K$ is the field of fractions of $R$.
    (a) Let $R=\mathbb{Z}[t]$, the ring of polynomials over $\mathbb{Z}$ in one variable. Let $p(x)=x^{n}+r_{n-1} x^{n-1}+ \ldots+r_{0} \in R[x]$ be a monic polynomial with coefficients in $R$, and suppose that $p(\alpha)=0$ for some $\alpha \in K$. Prove that $\alpha \in R$.
    (b) Now let $R=\mathbb{Z}[\sqrt{-3}]$. Find a monic polynomial $p(x) \in R[x]$ which has a root in $K$, but has no root in $R$ (and prove that $p(x)$ has required properties). Hint: There actually exists a quadratic polynomial with integer coefficients with required property.

29. (Jan 14 #2) Find all ring homomorphisms
    (a) from $\mathbb{Z}$ to $\mathbb{Z} / 30 \mathbb{Z}$;
    (b) from $\mathbb{Z} / 30 \mathbb{Z}$ to $\mathbb{Z}$.

30. (Jan $14 \# 3$ ) Let $\mathbb{Q}(\sqrt{-2})$ be a quadratic field with associated ring of integers $\mathcal{O}=\mathbb{Z}[\sqrt{-2}]$. Prove that $\mathcal{O}$ is a Euclidean Domain . (Hint: use the field norm.)

31. (Jan 14 #4) Prove that if $R$ is a principle ideal domain (P.I.D.) and $D$ is a multiplicatively closed subset of $R$ with $0 \notin D$, then $D^{-1} R$ is also a P.I.D.

32. (Aug $15 \# 1$ ) Find all maximal ideals of $\mathbb{Z}[i]$ which contain 182 . Find minimal generators for these ideals.

33. (Aug $16 \# 6$ ) (a) Name two examples of each of the following: (i) PIDs which are not fields
    (ii) UFDs which are not PIDs (iii) commutative integral domains which are not UFDs and
    (iv) commutative rings which are not integral domains, (v) noncommutative rings.
    (b) Given an example of a non-principal ideal in one of the examples you listed (with proof).

34. (Aug 17 #4) Let $R$ be a commutative ring with 1 which is Artinian, i.e. for any descending chain $I_{1} \supseteq I_{2} \ldots \supseteq I_{n} \ldots$ of ideals of $R$ there exists an $n_{0}$ such that $I_{n}=I_{n_{0}}$ for all $n \geq n_{0}$. Prove the following:
    (a) If $R$ is an integral domain, then it is a field.
    (b) Any prime ideal of $R$ is maximal.
    (c) $R$ has only finitely many maximal ideals.

35. (Aug $18 \# 3$ ) Let $R$ be a commutative ring such that all prime ideals are finitely generated. Prove that $R$ is Noetherian by completing the following steps.
    (a) Let $X$ be the set of non-finitely generated ideals of $R$. Prove that if $R$ is not Noetherian then $X$ has a maximal element, say $I$.
    (b) Prove that there exist elements $x, y \in R$ such that $x, y \notin I$ but such that $x y \in I$.
    (c) Prove that $I_{x}=I+R x$ and $J_{x}=\{r \in R \mid r x \in I\}$ are finitely generated ideals. Here, $x$ is the same element as in the previous part.
    (d) Conclude that $I$ is finitely generated and derive a contradiction.

36. (Aug $19 \# 3$ ) Let $m$ and $n$ be positive integers with $m \mid n$, and let $f: \mathbb{Z} / n \mathbb{Z} \rightarrow \mathbb{Z} / m \mathbb{Z}$ be the natural projection. Prove that the associated map of the groups of units $f:(\mathbb{Z} / n \mathbb{Z})^{\times} \rightarrow (\mathbb{Z} / m \mathbb{Z})^{\times}$is surjective.

37. (Aug $19 \# 6$ ) Let $R=\mathbb{Z}[x]$.
    (a) Let $I$ be an ideal of $R$ which contains a MONIC polynomial of degree $n$, call it $p(x)$. Prove that $I$ can be generated (as an ideal) by at most $n+1$ elements. Hint: Consider the quotient $I /(p(x))$.
    (b) Now let $M=(2, x)$ and $I=M^{2}=\left(4,2 x, x^{2}\right)$. Prove that $I$ cannot be generated (as an ideal) by less than 3 elements. Hint: Consider the quotient $M^{2} / M^{3}$.

38. (Jan $20 \# 3$ ) Let $R=\mathbb{Z}[i]$, the ring of Gaussian integers. Find (with complete proof!) the number of ideals of $R$ which contain 30 .

39. (Jan 20 #4) Let $R$ be a commutative ring with 1 and let $x \in R$.
    (a) Prove that $x \notin R^{\times}$if and only if $x \in M$ for some maximal ideal $M$ of $R$.
    (b) Prove that the following are equivalent:
    (i) $x \in M$ for every maximal ideal $M$ of $R$
    (ii) $1+x y \in R^{\times}$for every $y \in R$

40. (Aug $20 \# 3$ ) Let $R=\mathbb{Z}[\sqrt{3}]$. You may assume without proof that $R$ is a Euclidean domain. Let $p \in \mathbb{N}$ be a prime number with $p \equiv 5 \bmod 12$. Prove that $p$ is a prime element of $R$.
    Hint: $12=3 \cdot 4$.

41. (Aug $20 \# 4$ ) Let $R$ be a commutative ring with 1 . An ideal $I$ of $R$ is called irreducible if $I$ cannot be written as $I=J \cap K$ where $J$ and $K$ are both ideals strictly containing $I$.
    (a) Prove that every prime ideal is irreducible.
    (b) Assume that $R$ contains a nonzero nilpotent element (that is, a nonzero element $a$ such that $a^{n}=0$ for some $n$ ). Prove that $R$ contains an irreducible ideal which is not prime. Hint: Fix a nonzero nilpotent element $a \in R$. Consider the set of all ideals NOT containing $a$ and show that this set has a maximal element.

42. (Jan $21 \# 3$ ) Let $R$ be a commutative ring with 1 , and assume that $|R|>1$. Prove that $R$ has at least one minimal prime ideal. Make sure to include all the details.
    Hint: Use Zorn’s lemma "backwards".
    Note: A prime ideal $P$ of $R$ is called a minimal prime ideal if $R$ has no prime ideals strictly contained in $P$. In particular, the zero ideal is a minimal prime whenever it is prime.

43. (Aug 21 #4) Let $R=\mathbb{Z}[\sqrt{-11}]=\{a+b \sqrt{-11}: a, b \in \mathbb{Z}\} \subset \mathbb{C}$, and let $I=(3,1+\sqrt{-11})$ be the ideal of $R$ generated by 3 and $1+\sqrt{-11}$.
    
    (1) Prove that $I$ is maximal.
    
    (2) Prove that $I$ is not principal.

Here are some further standard facts/problems concerning rings of fractions ( $S$ always denotes a multiplicatively closed subset of $R$ containing 1 ).
44. Show that the prime ideals of $S^{-1} R$ are in one-to-one correspondence with the prime ideals of $R$ not containing any element of $S$. Is a similar statement true for maximal ideals?
45. Show that $S^{-1} R$ is a local ring (i.e. has precisely one maximal ideal) if $S=R \backslash P$ for a prime ideal $P$ of $R$. In this case we use the notation $R_{P}:=S^{-1} R$.
46. Prove that $R$ does not have any nonzero nilpotent elements if and only if, for all prime ideals $P, R_{P}$ does not have any nilpotent elements.
47. Suppose that $S=\left\{a^{n} \mid n \in \mathbb{N}_{0}\right\}\left(a^{0}:=1\right)$ for some $a \in R \backslash\{0\}$. Show that $S^{-1} R$ is isomorphic to $R[x] /(a x-1)$. What does this mean for nilpotent $a$ ?