3
votes
2answers
56 views

Prove $M$ is a Maximal Ideal in $\Bbb Z\times \Bbb Z$

A problem from introduction to abstract algebra by Hungerford. It asks: If $p$ is a prime integer, prove that $M$ is a maximal ideal in $\mathbb Z \times \mathbb Z$, where $M =\{(pa,b)\mid a,b\in ...
1
vote
1answer
49 views

Necessary and sufficient condition for $r(\mathfrak a)$ to be prime

As we know, $$\mathfrak a~\text{is a primary ideal}\Rightarrow r(\mathfrak a)~\text{is a prime ideal}. $$ But $r(\mathfrak a)$ may not be a prime ideal if $\mathfrak a$ isn't a primary ideal. ...
1
vote
2answers
55 views

Checking the maximality of an ideal

Let $R = \mathbb{Z}_{(2)}$ be the localization of $\mathbb{Z}$ at the prime ideal generated by $2$ in $\mathbb{Z}$. Then prove that the ideal generated by $(2x-1)$ is maximal in $R[x]$. Otherwise ...
0
votes
1answer
19 views

Find all elements of quotient ring

I am studying the definitions of rings, ideals, and quotient ring, but I have a bit problem to apply the theory into the practice. I would like to find all elements of quotient ring $\mathbb{Z}[i]/I ...
1
vote
0answers
37 views

Question about Principal Ideals

I'm just learning basic ring theory and had a question about the definition of a principal ideal. For a commutative ring $R$ with unity, Fraleigh defines the principal ideal generated by $a\in R$ as ...
1
vote
1answer
24 views

Show $I=p\mathbb{Z}$ for prime $p$.

Let $I\subset\mathbb{Z}$ be an ideal such that $I\neq \mathbb{Z}$ and if $I\subset J\subset\mathbb{Z}$ then $I=J$ or $J=\mathbb{Z}$. Show that $I=p\mathbb{Z}$ for some prime $p$. Attempt: We know ...
0
votes
0answers
15 views

Explanation why $R_P=(S^{-1}R)_{P(S^{-1}R)}$

Suppose $R$ is a ring and $P$ a prime ideal. If $S$ is a mutliplicative subset, can anyone explain why we have the equality $R_P=(S^{-1}R)_{P(S^{-1}R)}$ when seen as subsets of the quotient field of ...
1
vote
1answer
26 views

Free modules and ideals

I am trying to show that an ideal I of R=$\mathbb{C}[x_1,x_2]$ generated by $x_1, x_2$ is free R-module. I am trying to show that I has a basis of the two generators given above. But I am not able to ...
-1
votes
1answer
76 views

Number of maximal and prime ideals

Find how many prime and maximal ideals there are in the ring consisting of matrices $$M= \begin{bmatrix} a & b & c \\ 0 & a & b \\ 0 & 0 & a \\ \end{bmatrix} $$ ...
3
votes
2answers
79 views

Is quotient of a ring by a power of a maximal ideal local?

Say I have a commutative ring $R$ with a maximal ideal $m$. Then $m/m^k$ is a maximal ideal in $R/m^k$ for any $k$. Is it the only maximal ideal, i.e. is $R/m^k$ a local ring? This is a well ...
1
vote
1answer
18 views

Does $I(J\cap K)=IJ\cap IK$ hold in a Dedekind ring?

For ideals in any ring, we have the relation $I(J\cap K)\subseteq IJ\cap IK$. Do we actually have equality if we are in a Dedekind domain? I've been looking around for a reference, but haven't found ...
3
votes
1answer
70 views

Confused on a proof that $\langle X,1-Y\rangle$ is not principal

I'm getting stuck on a passage in my notes. The claim is that the ideal $P=\langle X,1-Y\rangle$ is not principal in $\mathbb{Q}[X,Y]/\langle 1-X^2-Y^2\rangle$. This follows since $P^2=\langle ...
0
votes
2answers
73 views

Tensor product of quotient rings [duplicate]

$A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?
2
votes
1answer
45 views

$A_{p}$ is a field when $p$ is a minimal prime and $A$ reduced

$A$ is a reduced commutative ring with unit; $p$ is a minimal prime ideal. If $S = A \setminus{p}$ , I have to show that the ring $A_{p} = S^{-1}A$ is a field. My thoughts: Since $p$ is a minimal ...
1
vote
0answers
30 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where p is prime integer and f is primitive integer polynomial that is irreducible ...
0
votes
1answer
27 views

Radical of an ideal - prove every prime ideal that contains $I$ also contains $\sqrt{I}$

Let $R$ be a commutative ring with a unit and $I$ an ideal. Please prove that every prime ideal that contains $I$ also contains $\sqrt{I}$. I easily conclud that $I \subseteq \sqrt I$ but I ...
-1
votes
1answer
65 views

Maximal multiplicative set and minimal prime ideal

Let $A$ be a ring and $P$ a prime ideal included in $A$. Show that $A \setminus P$ is a maximal multiplicative set if and only if $P$ is a minimal prime ideal of $A$. What can be the proof for this ...
0
votes
1answer
27 views

Minimal prime ideals consist of zerodivisors [duplicate]

I don't find the proof for this little demonstration ... Let $P$ be a minimal prime ideal of $A$. Show that $P$ is contained in the set of zero divisors of $A$.
0
votes
1answer
56 views

Invertible elements and maximal ideals of a localization

Let $n\in\mathbb Z$ and let $A$ be the set of integers co-prime to $n$. Denote $A^{-1}\mathbb Z$ by $\mathbb Z_{(n)}$. 1) Find the invertible elements of $\mathbb Z_{(6)}$ My attempt: let $m$ be ...
0
votes
1answer
33 views

Question about comaximal ideal proof

Let $A$ be a ring and $M\subseteq A$ a maximal ideal. Show that if $I\subseteq A$ such that $I\not\subseteq M$, then $M$ and $I$ are comaximal($M+I=A$). I cannot find the proof for this statement.
0
votes
1answer
21 views

Show that these rings of Gaussian integers are ideals in $\mathbb{Z}[i]$?

Consider the ring of Gaussian integers: $\mathbb{Z}[i]$ = {a + bi | a, b ∈ Z} ⊂ $\mathbb{Q}[i]$ with $i^2$ = −1. Let I = $(2+3i)$ and J =$(2−3i)$. Show that I and J are ideals of $\mathbb{Z}[i]$.
2
votes
1answer
53 views

Maximal ideals in $\mathbb{Z}[x]$

I am trying to solve the following problem from Artin: Every maximal ideal $\mathbb{Z}[x]$is of the form $(p,f)$ where p is a prime integer and $f$ is a primitive polynomial that is irreducible modulo ...
0
votes
1answer
36 views

Ideals in direct product of rings

I am trying to solve that problem: Let $ R_1,...,R_n$ rings with identity, every ideal of $R=\prod_{i=1}^n R_i$ is in form $\prod_{i=1}^n I_i$ where $ I_i$ideal of $R_i$. The first part is clearly if ...
0
votes
2answers
34 views

Spec($A$) is connected if $A$ is local

Another exercise from Balwant-Singh: Show that if $A$ is local then Spec($A$) is connected in the Zariski topology. Any hint ?
1
vote
1answer
42 views

Idempotent/Spec

I'm studying Basic Commutative Algebra by Balwant-Singh; I'm stuck on this exercise: $A$ is a commutative ring; show this $3$ conditions are equivalent: 1) $A$ contains a non-trivial idempotent 2) ...
1
vote
1answer
32 views

When an Intersection of Prime Ideals is a Prime Ideal

Let $R$ be an arbitrary ring, $\{P_1,....,P_n\}$ be a set of prime ideals. Verify that $P_1 \cap ... \cap P_n$ is prime if and only if there exists $1 \leq i \leq n$ such that $P_i$ is contained in ...
1
vote
2answers
39 views

Prime ideal in a ring with some property is maximal

Let $R$ be a ring where for every element $a \in R$, there exists a positve integer $n_a \gt2$ such that $a^{n_a}=a$. Prove that every prime ideal in $R$ is maximal. I think that I would want to ...
0
votes
1answer
58 views

Direct product of algebras over a field

Let $ B_1,B_2,...,B_n$ k-algegras, $ B=\prod_{i=1}^{n}B_i $ the direct product of those (k is a field) , and $ J_i$ an ideal of its k-algebra. i must to prove that: The direct product $ ...
1
vote
1answer
72 views

Give an example of an ideal in $\mathbb{Z}\times\mathbb{Z}$ which is maximal.

My answer right now is just $(0,1)$ and $(1,0)$ resulting in $\mathbb{Z}\times\mathbb{Z}$ as $(1,1)$. But this is the entire ring... Help?
1
vote
1answer
43 views

Prime ideals in formal power series

Let $A$ be a commutative ring with unit. If $\mathfrak{p} \subset A $ is a prime ideal, then $\mathfrak{p}$ is the contraction of a prime ideal of $A[[x]]$, the ring of formal power series. Why is ...
-2
votes
3answers
100 views

Ring theory question: $I=\langle x,2 \rangle$ prime/maximal ideal in $\mathbb Z[x]$?

In $\mathbb{Z}[x]$ , let $I = \lbrace f(x) \in \mathbb{Z}[x] : f (0) \text{ is an even integer} \rbrace.$ Is $I=\langle x,2 \rangle$ a prime ideal of $\mathbb{Z}[x]$? Is $I=\langle x,2 \rangle$ a ...
7
votes
1answer
113 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
2
votes
1answer
86 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
1
vote
2answers
36 views

In $\mathbb{Z}[t]$, $Q = (4, t)$ is not a power of $M = (2, t)$

The problem of showing that Q, as above, is not a power of M, as above, rises as part of a larger problem. I'm confident about my response to the other parts, but the best justification I can come up ...
0
votes
1answer
30 views

Finitely generated ideal in boolean ring [duplicate]

A boolean ring is a commutative ring where $x^{2} = x$ for every $x$. Why in such a ring a finitely generated ideal is principal ?
2
votes
1answer
38 views

The ideal $I=\{a_{0}+a_{1}x+\cdots+a_{k}x^{k} \in F[x]\mid a_{0}+a_{1}+\cdots+a_{k}=0\}$ equals $\langle x-1\rangle$?

If $I=\{a_{0}+a_{1}x+\cdots+a_{k}x^{k} \in F[x]\mid a_{0}+a_{1}+\cdots+a_{k}=0\}$ is an ideal in $F[x]$, is it equivalent to $I=\langle x-1\rangle$? Think it is because $1$ is obviously a root of ...
4
votes
3answers
199 views

Idempotent in a local ring

Is it true that a local ring, i.e. a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
2
votes
1answer
40 views

Height of a specific maximal ideal

Let $k$ be a field, $k[x,y^2,xy,y^3]$ our ring and $\mathfrak a$ the ideal generated by $x,y^2, xy,y^3$. I want to determine the height $h(\mathfrak a)$ of $\mathfrak a$. My ideas: We see easily ...
1
vote
2answers
65 views

Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
1
vote
1answer
52 views

Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
2
votes
1answer
45 views

Let $k$ be a division ring, then the ring of upper triangular matrixes over $k$ is hereditary

I'm reading Ring Theory by Louis H. Rowen, and he claimed that The ring of upper triangular matrices over a division ring is hereditary (it's on page 196, Example 2.8.13 of the book). I think it ...
2
votes
1answer
76 views

$\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = \{ \mathfrak{p}A_\mathfrak{p}\} $

Let $k$ be a field, $A = k[X_1,X_2,...]$, $\mathfrak{p} = (X_1,X_2,...)$, $I = (X_1^2-X_1,X_2^2-X_2,...)$, $M= A/I$. I am trying to show that $\operatorname{Ass}_{A_\mathfrak{p}}(M_\mathfrak{p}) = ...
1
vote
1answer
50 views

Polynomial rings over a field and maximal/prime ideals

Let $F$ be a field , I want to prove that every proper nontrivial prime ideal of $F[x]$ is maximal. My definitions of prime/maximal ideals are as follows: $N$ is a prime ideal of $R$ iff $ab \in N ...
2
votes
1answer
39 views

A non-zero and non-invertible element in a noetherian integral domain has a decomposition into irreducible elements

Let $R$ be a noetherian integral domain. I want to show that any non-zero and non-invertible element $a$ can be written as a finite product of irreducible elements. my ideas: I should argue by ...
2
votes
1answer
36 views

$\dim (A/I) \le \dim (A)$

Let $A$ be a ring and $I$ be an ideal. I'm trying to prove that $\dim (A/I) \le \dim (A)$. My attempt to proof Suppose that $\dim (A)=n$, then there are prime ideals $\mathfrak ...
0
votes
1answer
41 views

Maximal and prime ideals of $\mathbb{Z} \times \mathbb{Z}$

I have to find a maximal ideal of $\mathbb{Z} \times \mathbb{Z}$ , and a prime ideal that is NOT maximal. Or, essentially, I want $I$ such that $\mathbb{Z} \times \mathbb{Z} / I$ is a field, and I ...
0
votes
1answer
35 views

Problems with a ring isomorphism

Let $k$ be a field and consider $a=(a_0,\ldots,a_n)\in k^{n+1}$ with $a_0\neq0$. Now $\rho(a)=\left(\{a_iT_j-a_jT_i\;:\; 0\le i<j\le n\}\right)$ is an homogeneous ideal of $k[T_0,\ldots,T_n]$ and I ...
2
votes
1answer
54 views

Algebraic Geometry and Maximal ideals

I am solving the following problem but couldn't figure out a strategy to solve: Does $(x^3-17, y^2)$ generate maximal ideals in the quotient ring $R=\mathbb{C}[x,y]/I$ where $I$ is the principal ...
1
vote
1answer
41 views

Dfference between strongly prime and prime ideal

An ideal $P\subset R$ is strongly prime, if for any $x$ and $y$ in the quotient field of $R$, $xy\in P$ implies $x\in P$ or $y\in P$. What is the difference between strongly prime ideal of $R$ and a ...
1
vote
1answer
31 views

The sum of all right ideals isomorphic as modules to a simple module is an ideal

I could use some help on the following problem. Let R be a ring. (a) If $r \in R$ and $U$ is a minimal right ideal of $R$, show that either $rU=0$, or that $rU$ and $U$ are isomorphic right ...