# Tagged Questions

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### 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 $\Bbb R[x,y]/J \cong \Bbb C$.
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### 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 ...
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### 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). ...
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### 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 ...
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### 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 ...
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### 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 $\mathbb{C}[x,y,z]$.
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### 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 ...
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### 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 ...
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### 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 $P$. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $R$.
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### 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 $a$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $(t-a)^n$. ...
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### 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 + ...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
### 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 ...
### 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 ...