0
votes
2answers
50 views

Maximal ideal which isn't principal

Let $J=(x-2,x-y^2-3)$ ideal in the polynomial ring $\Bbb R[x,y]$. Please help me prove that $J$ is a maximal ideal which isn't principal, and that $\Bbb R[x,y]/J \cong \Bbb C$.
1
vote
0answers
65 views

Ideals of $\Bbb Z/p^2q\Bbb Z$

Let $p,q$ be distinct primes. Then $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct prime ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 2 distinct prime ...
0
votes
0answers
30 views

Relation between the initial ideal and radical

Let $I$ be an ideal of the polynomial ring $S$. Show that ${\rm In}(\sqrt I)\subseteq\sqrt{{\rm In}(I)}$, where by ${\rm In}(I)$ we denote the ideal of initial forms of I, In(I) = (In(f) : f $\in$ I). ...
1
vote
1answer
31 views

Lie group and generated ideals

I have this question in my textbook, and I can't seem to solve it on my own: Let $P \subset GL(n,\mathbb{C})$ be a subgroup as following: $P$ consists of all matrices in block ...
3
votes
1answer
25 views

Ideal in Dedekind domain

Let $D$ be Dedekind domain and $I$ nonempty ideal in $D$. I have to show that there are only finitely many ideals $J$ in $D$ such that $I$ is contained in $J$. My first idea would be: assume that ...
-4
votes
1answer
113 views

An ideal which is not maximal in $\mathbb{C}[x,y,z]$

Show that $$J=(x^2+y^2+z^2+x+y+z, x^5+y^5+z^5+2(x+y+z), x^7+y^7+z^7+3(x+y+z))$$ is not the maximal ideal $m=(x,y,z)$ in $\mathbb{C}[x,y,z]$.
0
votes
1answer
29 views

How to find the ideals of $\Bbb{Z}_n$

I have a homework problem to find the maximal ideals in $\Bbb{Z}_8$, $\Bbb{Z}_{10}$, $\Bbb{Z}_{12}$, and $\Bbb{Z}_n$. That question has already been asked on here, but I don't even understand how to ...
2
votes
1answer
42 views

Isomorphism between a sub-ring to $\Bbb Q$, the rational field

Let $R$ be a commutative ring with an identity, which contains $\Bbb Z$, the integer field, as a sub-ring with the same identity element. Let $I$ be a maximal ideal in $R$. I need to prove that if ...
0
votes
0answers
25 views

Ireland and Rosen, question 12.28 (unique factorization in prime ideals)

I'm stuck with the following exercise in Ireland and Rosen, chapter 12. Let $D$ be the ring of integers in a number field $F$. Suppose $(p)=P^2A$ for $p$ prime in $\Bbb Z$ and a prime ideal $P$. ...
2
votes
2answers
55 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
1
vote
2answers
22 views

Size of an ideal in a polynomial Ring

Let $F$ be a field and let $I = \{f(x) \in F[x]\mid f(a) = 0 ~~ \forall a \in F\}$. Prove that $I = \{0\}$ when $F$ is infinite. I have already shown that $I$ is an ideal and that $I$ is infinite ...
0
votes
1answer
48 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
vote
0answers
55 views

Class Group of Ring of Integers of $\mathbb{Q}[\sqrt{-57}]$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need ...
1
vote
1answer
19 views

Maximal element of $(I : x)$, where $x$ is in $A - I$, is prime belonging to $I$

Given that $I$ is decomposable, I am supposed to prove that any maximal element $P$ of the set {$(I : x) | x \in A - I$} must belong to $I$, i.e., $P$ is prime and for every reduced primary ...
1
vote
2answers
37 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 ...
2
votes
1answer
84 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
89 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 ...
0
votes
1answer
62 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
2answers
37 views

How does one find a minimal primary decomposition?

What exactly does it mean for a primary decomposition to be "minimal" and is the a general method to obtain such decompositions? I've tried looking at some examples but they all give very little ...
0
votes
2answers
33 views

Common divisors in a PID

Suppose that R is an integral domain and that α, β, γ ∈ R. We say that γ is a common divisor of α and β if γ|α in R and γ|β in R. Suppose that R is a PID. Suppose that α, β ∈ R. Let I = (α) and ...
1
vote
2answers
191 views

Prove: The pre-image of an ideal is an ideal.

Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$.
1
vote
1answer
45 views

Principal prime ideal is generated by irreducible element

$R$ is an integral domain, $x\in R$ and $(x)=I$ is a prime ideal. Prove that $x$ is an irreducible element of $R$. So I assume $ab\in I$, with $a, b \in R$. Since $I$ is a prime ideal, either $a$ ...
5
votes
1answer
70 views

Ideal contained in the union of two ideals and a prime

Taken from Miles Reid "Undergraduate Commutative Algebra" p.35 ex. 1.12 b) Let $I,J_1,J_2 \subset A$ be ideals of a commutative ring $A$. Let $P$ be a prime ideal, then if $I \subset J_1 \cup J_2 ...
5
votes
1answer
73 views

Find a maximal ideal $I$ in the ring $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/I$ is isomorphic to $\mathbb{Z}/521\mathbb{Z}$.

I know $\mathbb{Z}[i]$, the Gaussian integers, is a PID. So $I$ is generated by a single element. At first I thought $I=(521)$, but $521$ can be reduced to $11^2 + 20^2$. Would $I=(11 + 20i)$ or ...
1
vote
1answer
76 views

Finish a proof that every prime ideal of a ring is the contraction of a prime ideal in its formal power series

Given a commutative ring $A$ with identity, and its formal power series ring $A[[x]]$, I am attempting to prove that every prime ideal of $A$ is the contraction of a prime ideal of $A[[x]]$. ...
3
votes
1answer
52 views

Maximal ideal not containg the set of powers of an element is prime

In the midst of attempting to prove that for a commutative ring $A$ with identity, and an ideal $I$ of $A$, $I = rad(I)$, where $rad(I) = \{x: x^m \epsilon I, m >0\}$, implies that $I$ is an ...
0
votes
2answers
92 views

Why $I=\left\{p(x)\in \mathbb{Z}\left[X\right]:2\mid p(0)\right\}$ is not a principal ideal? [duplicate]

I saw this question but I still do not understand: What is the difference between ideal and principal ideal? At my homework I had to prove to things about $I=\left\{p(x)\in ...
3
votes
1answer
110 views

Ideals containing $(6, x^3-1)$ in $\mathbb{Z}[x]$

I want to find all ideals containing $(6, x^3-1)$ in $\mathbb{Z}[x]$ and I can only find ten ideals: $\mathbb{Z}[x]$ $(2, x-1),\; (2, x^2+x+1),\; (3, x-1)$ $(6,x-1),\; (2,x^3-1),\; (6, x^2+1+1),\; ...
0
votes
1answer
50 views

Qustion about Ideal…(Ring theory)

I know that if $I,J$ are Ideals of $R$ so $I+J=\left\{i+j|i\in I, j\in J\right\}$ is Ideal to... I need to find $a\in \mathbb{Z}$ s.t. ...
1
vote
0answers
86 views

Check whether an ideal is maximal or prime

Problem. Check whether the following ideals are maximal or prime in $\mathbb{Z}[X_1,X_2]$ and $\mathbb{Q}[X_1,X_2]$: i) $(X_1,X_2)$ ii) $(X_1+X_2)$ iii) $(X_1,X_2,2)$ iv) ...
1
vote
1answer
342 views

Prime/maximal ideals of $\mathbb{C}[x, y]$ containing a given ideal

Remember that (i) every maximal ideal is a prime ideal, (ii) for proper ideals $I$ of rings $R$, the factor ring $R/I$ is a field iff $I$ is a maximal ideal of $R$, and that (iii) whenever $F$ (for ...
1
vote
1answer
181 views

Showing an ideal is not equal to $\Bbb Z[i]$

Let $p = 4m + 1$ and $t = (2m)!$. Consider the ideal $I = (p, t + i)$ of $\Bbb Z[i]$ generated by $p$ and $t + i$. Show that $I$ is not equal to $\Bbb Z[i]$. I started by trying to show that I is ...
2
votes
2answers
115 views

Finding all ideals in $\mathbb{C}[[x]]$

I am currently trying to find all ideals in $\mathbb{C}[[x]] = \{\sum_{i=0}^\infty a_ix^i : a_i \in \mathbb{C}\}$, that is, the ring of Taylor series with complex coefficients. I know that since the ...
2
votes
4answers
53 views

Consider the ideal $\langle 1+i \rangle$ in $\mathbb{Z}[i]$.

Consider the ideal $\langle 1+i \rangle$ in $\mathbb{Z}[i]$. $(a)$ Make use of the given description of this ideal, $\hspace{75pt}$ $\langle 1+i \rangle = \{a+bi:a+b \text{ is even}\}=\{\alpha\in ...
0
votes
2answers
57 views

Are these ideals the same?

I have already proved that $(X^3-Y^3,X^2Y-X)\subseteq(X^2-Y,X-Y^2)$ since the elements $X^3-Y^3$ and $X^2Y-X $ can be written as a linear combination of $(X^2-Y,X-Y^2)$. However, I can't write ...
1
vote
2answers
74 views

Ring homomorphism homework including ideals and surjectivity.

$R$ is a ring and $I$ and $J$ are ideals of $r$. Show that the ring homomorphism $h:R \rightarrow R/I \times R/J, r \mapsto (r+I,r+J)$ is surjective iff $I+J=R$ give a description of the kernel of ...
4
votes
2answers
121 views

Show that quotient ring of a $\Bbb C$-algebra by a maximal ideal is isomorphic to $\mathbb{C}$.

Let $R = \mathbb{C}[x_1,...,x_n]/I$ be a quotient of a polynomial ring over $\mathbb{C}$, and let $M$ be a maximal ideal of $R$. How do I show that quotient ring $R/M$ is isomorphic to ...
2
votes
1answer
70 views

Prove that the elements $1, t-a, (t-a)^2, (t-a)^3,\dots, (t-a)^{n-1}$ form a $\mathbb{C}$-basis for the quotient ring $\mathbb{C}[t]/((t-a)^n)$.

Prove that the elements $1, t-a, (t-a)^2, (t-a)^3,\dots, (t-a)^{n-1}$ form a $\mathbb{C}$-basis for the quotient ring $\mathbb{C}[t]/((t-a)^n)$. $((t-a)^n)$ is the ideal generated by $(t-a)^n$. ...
2
votes
4answers
320 views

Example of a commutative ring with identity with two ideals whose product is not equal to their intersection

I need a specific example of a commutative ring with identity, and two ideals in the ring whose product is not equal to their intersection. I know that for two such ideals I and J, IJ = I ∩ J if I + ...
3
votes
1answer
64 views

Showing that the ideal of matrices with entries from an ideal is maximal iff that ideal is maximal

I know that if $J$ is an ideal in a ring $R$ then $M_n(J)$, the set of all $n\times n$ matrices with entries in $J$, is also an ideal. How would I show that $M_n(J)$ is maximal iff $J$ is maximal? I ...
3
votes
3answers
140 views

Maximal ideal in the ring of functions from $\mathbb{R} \to \mathbb{R}$

Well, the problem I'm trying to solve is this: Let A be the ring of all continuos functions from $\mathbb{R} \to \mathbb{R}$. Show that $$M = \{f \in A; f(0)=0\}$$ is a maximal ideal of A. I tried ...
3
votes
0answers
57 views

About quotient ring

I want to find the value $|\mathbb{Z}({\sqrt{2})/(3+\sqrt{2})}|,|\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}|$ and also the number of ideals of $\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}$. But still not ...
2
votes
1answer
55 views

Some homework about ideals

I will ask you some things about ideals. Determine weither the ideal $(X^2+3) \subset \mathbb{F}_5[X]$ is maximal or prime. Intuitively I'd say that the ideal is prime but not maximal. To prove ...
4
votes
1answer
31 views

Semigroup which satisfied $(xy)^{\pi} = (xy)^{\pi}x$ and principal right ideals

Let $(S,\cdot)$ be a finite semigroup, then every $s \in S$ has an unique idempotent power, i.e. there exists a smallest $i \in \mathbb N$ such that $s^i$ is idempotent. It is the unique idempotent in ...
3
votes
1answer
91 views

Product of Ideals

Can I define the product of ideals $I, J$ as $$IJ = \left\{\sum_{j}^n \sum_{i}^n a_i b_j| a_i \in I, b_j \in J, i,j \in \mathbb{N}\right\}.$$ Are there some books that define the product of ideals ...
2
votes
2answers
545 views

Maximal ideals in the ring of real functions on $[0,1]$

Assume $S$ to be all continuous functions from $[0,1]$ to $\mathbb R$. How to prove that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$? Thanks in advance.
4
votes
2answers
93 views

Proving a factorization of ideals in a Dedekind Domain

Let $R=\mathbb{Z}[\sqrt{-13}]$. Let $p$ be a prime integer, $p\neq 2,13$ and suppose that $p$ divides an integer of the form $a^2+13b^2$, where $a$ and $b$ are in $\mathbb{Z}$ and are coprime. Let ...
3
votes
2answers
144 views

Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$

I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$. I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
4
votes
1answer
168 views

Solving $x^2+19=y^5$

I was given several exercises and there is a particular one, I am not able to solve. Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...
2
votes
1answer
73 views

ideals in $C^*$ algebra

Let $A$ be a $C^*$ algebra and $I$ be a closed ideal in $A$. Prove that for all $a\in A$, $a\in I$ iff $a^*a\in I$. I want to prove that if $a^*a\in I$, then $a\in I$, and I know the following fact ...