An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ...

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How to find all ideals of this factor ring?

My question is about how to find all ideals of a factor ring. Let $J=((x+1)(x+2)(x+3)) \subset\mathbb C[x]$ and let $R=\mathbb C[x]/J$. I want to find all ideals in $R$ that contain $J$. My ...
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96 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) ...
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1answer
198 views

How to check if an ideal generated by two elements is prime?

I have a question about ideals generated by two elements. I've searched MathStackexchange and found some related posts, but I haven't been able to understand how it all works. The question is in ...
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62 views

Why $\langle I, J\rangle =R$ for distinct prime ideals $I$, $J$ of a principal ideal domain $R$?

Let $R$ be a principal ideal domain with identity and $I$, $J$ be distinct prime ideals of $R$. Prove that $1 \in \langle I, J\rangle$ hence $\langle I, J\rangle = R$. How to prove?
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59 views

Any prime ideal of $R[x]$ properly containing $M[x]$ is a maximal ideal of $R[x]$ [duplicate]

Let $M$ be a maximal ideal in a ring $R$. Prove that any prime ideal of $R[x]$ properly containing $M[x]$ is a maximal ideal of $R[x]$. Help me some hints
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1answer
185 views

Let $M$ be a maximal ideal of a ring $R$. Is $M[x]$ a maximal ideal of $R[x]$?

Let $M$ be a maximal ideal of a ring $R$. Prove that $M[x]$ is not a maximal ideal of $R[x]$ Thanks a lot
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72 views

Prove that $R\left[x\right]/I\left[x\right]\cong\left(R/I\right)\left[x\right]$ [duplicate]

Let $I$ be an ideal of a ring $R$, define $I[x]$ to be the set of all polynomials whose coefficients are in $I$. Prove that $R\left[x\right]/I\left[x\right]\cong\left(R/I\right)\left[x\right]$ Help ...
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1answer
430 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 ...
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1answer
58 views

Are two different prime ideals relatively prime?

Are two different prime ideals relatively prime? Thanks in advance!
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1answer
68 views

Find the maximal ideals of the ring $\mathbb{Z}_{36}$.

Find the maximal ideals of the ring $\mathbb{Z}_{36}$. I don't know where to start on this one. Any help/hints would be greatly appreciated.
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1answer
72 views

Krull dimension and graded prime ideals

How can we show that $\dim R/p=0\Leftrightarrow p=(x_{1},\ldots,x_{n})\Leftrightarrow R/p\simeq\mathbb{K}$, where $R=\mathbb{K}[x_{1},\ldots,x_{n}]$ is considered graded with standard grading (i.e. ...
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1answer
97 views

isomorphism between factor ring of matrices and Z

I have a commutative ring R= $\begin{pmatrix}a & b \\ 0 & a \end{pmatrix}$ (R is a 2x2 matrix, a, b $\in$ Z), I=$\begin{pmatrix}0 & b \\ 0 & 0 \end{pmatrix}$ is an ideal. I need to ...
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1answer
248 views

Find all prime and maximal ideals of $\mathbb C[x,y]$ that contain $I=\langle x^2 + 1, y + 3\rangle$

I want to find all prime and maximal ideals of $\mathbb C[x,y]$ that contain $I=\langle x^2 + 1, y + 3\rangle$. My approach is that I know that if $f(x)$ is irreducible then $ < f(x) > $ is ...
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2answers
342 views

Maximal ideals in the ring of Gaussian integers

Let $R= \{ a+bi : a,b \in \mathbb{Z} \}$ be a subring of $\mathbb{C}$. Consider two principal ideals $I=(7)$ and $J=(13)$ in $R$. Is the ideal $I$ maximal? How about $J$? I don't understand what ...
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69 views

How is this map a well-defined homomorphism?

If $f: R \rightarrow S$ is a homomorphism of rings with kernel $K$, and $I$ is an ideal in $R$ such that $I \subset K$. The hypothesis is that the map $\overline{f}: R/I \rightarrow S$ given by ...
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3answers
103 views

Number of elements in $D/P^e$ where $D$ is a ring of algebraic integers, and $P$ a prime ideal

This is from Ireland and Rosen's A Classical Introduction to Modern Number Theory. Proposition 12.3.2: Consider a field $F/\mathbb Q$ with ring of integers $D$, and a prime ideal $P$ of $D$. Then ...
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1answer
80 views

How to show $a$ is an element of every maximal ideal of ring $R$ iff $1-ab$ is a unit for all $b \in R$?

Let $R$ be a commutative unitary ring. The task is to prove the statement which says that for an element $a\in R$ stands: $a$ is an element of every maximal ideal of $R$ iff $1-ab$ is a unit for all ...
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1answer
64 views

Zero ideal and domains

Let $R \neq 0$ be a commutative ring. I think we have $$ R \text{ is a domain } \iff (0) \text{ is a prime ideal of } R.$$ The argument is straightforward: let $a,b \in R$ such that $a\cdot b = 0$. ...
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Proving that the ring $R$ of $2\times 2$ matrices over $\Bbb{Q}$ contains only two ideals: $(0)$ and $R$. [duplicate]

There's a question in Herstein: Prove that the ring $R$ of $2\times 2$ matrices defined over $\Bbb{Q}$ contains only two ideals: $(0)$ and $R$. This seems to say that if I take any non-zero ...
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1answer
95 views

Every maximal ideal is prime… why not converse?

I know that every maximal ideal is prime but I don't see why the converse doesn't hold. Intuitively it seems like every prime ideal should be maximal. Off the top of my head I can't imagine how we ...
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1answer
67 views

Proving (by using Zorn's lemma) that every nonempty set contains a maximal ideal

I am trying to prove the following exercise: Let $X \neq \emptyset$. Prove, (by using Zorn's Lemma) that there exists a maximal ideal in $P(X)$. Proof: Take $\mathcal{J}$ to be the set of all ideals ...
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Is it true that in a Noetherian ring every descending chain of prime ideals stabilizes?

Is it true that in a Noetherian ring every descending chain of prime ideals stabilizes? It would be good if I had this result. As it would finish off my proof that the minimal primes of an ideal ...
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1answer
65 views

Krull dimension bound of a Fitting ideal

Given a finitely presented $R$-module $M$ over a ring $R$ one can define, for every integer $k\geq 0$ the $k$-th Fitting ideal of $M$, for instance in this way, using exterior algebra. ...
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$\mathbb Z\times\mathbb Z$ is principal but is not a PID

I need to find an example of a ring that is not a PID but every ideal is principal. I know that $\mathbb Z\times\mathbb Z$ is not an integral domain, so certainly is not a PID, but here every ideal is ...
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119 views

Some questions about Fitting ideals

Let $R$ be a ring and $M$ a finitely presented $R$-module. Given a free presentation $$ R^{\oplus m} \to R^{\oplus n} \to M \to 0 $$ we define $Fitt_k(M)$, the $k$-th Fitting ideal of $M$, to be the ...
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1answer
41 views

Generator for the ideal $I + J$ where $I = (2 + 3i)$ and $J = (1 - i)$

On a related question I calculated the GCD of $I = (2 + 3i)$ and $J = (1 - i)$ to be $1$. Now I know that $\mathbb{Z}[i]$ is a principal ideal domain. And I also know that the greatest common divisor ...
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If $P$ is a prime ideal of $R$, $\sqrt{P^{n}}=P\ \forall n\in\mathbb{N}$?

Let $R$ be a commutative ring. If $P$ is a prime ideal of $R$, $\sqrt{P^{n}}=P\ \forall n\in\mathbb{N}$?
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Ideals in a real/complex number field?

Considering a real or complex number field (with traditional addition and multiplication) I see no ideals besides $\mathbb{R}$ and $\{ 0\}$ or $\mathbb{C}$ and $\{ 0 + 0i\}$. Quick web search gave no ...
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$R$ has only one maximal ideal

Let $F$ be a field. Let $R$ be the set of all upper triangular matrices of the ring $M_{n}(F)$ with the property that its coefficients on the main diagonal are all the same. Prove that $R$ has only ...
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Is $\{0\}$ the only maximal ideal in $M_n(\mathbb{Q})$?

As the question, Is $\{0\}$ the only maximal ideal in $M_n(\mathbb{Q})$? Intuitively it should be true. I think that if there is a bigger ideal, then it should be $M_n(\mathbb{Q})$ itself but I ...
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1answer
36 views

proof of equivalent statements for an element of a ring belonging to every maximal left ideal of that ring

I would like to see a proof of the following. Let $R$ be a ring and let $a\in R$. Prove that the following conditions are equivalent. $a$ belongs to every maximal left ideal of $R$. $1+ra$ has ...
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1answer
48 views

Problem involving ideals contained in prime ideals [duplicate]

So far I have used the hint and have done the following: Let $i=1,2$. Suppose that $I\subseteq P_1\cup P_2$, but that $I$ is not properly contained in $P_i$ for any $i$. Then there is an $a_i$ such ...
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1answer
51 views

Let $R$ be an Euclidean domain. Let $p$ be irreducible. Let $a\ge1$. Show that every non-zero submodule of $R/Rp^a$ contains $Rp^{a-1}/Rp^a.$

Let $R$ be a Euclidean domain. Let $p$ be irreducible. Let $a\geq 1$. Show that every non-zero submodule of $R/Rp^a$ contains $Rp^{a-1}/Rp^a$. I bet the answer is stupidly obvious, but I just ...
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1answer
79 views

Ideals in commutative noetherian rings with unique prime ideal

Let $R$ be a commutative noetherian ring with $1$ having only one prime ideal $\mathfrak{P}$. It follows that $\mathfrak{P}^n = 0$ for some integer $n$. Can we say that every proper ideal in $R$ is a ...
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50 views

What some integer ideals of $\mathbb{Z}$ look like

Maybe I'm having a rough day, but I can't seem to wrap my head around what integer ideals of $\mathbb{Z}$ like $2\mathbb{Z}/12\mathbb{Z}$ or $3\mathbb{Z}/12\mathbb{Z}$ look like. I understand that ...
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440 views

Prime ideal in a polynomial ring over an integrally closed domain

Let $R$ be an integrally closed domain. Then $R[x]$ is also integrally closed. Let $P$ be a prime ideal of $R[x]$ with the property that $P\cap R=\left\{ 0\right\} $ and $P$ contains a monic ...
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1answer
185 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 ...
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1answer
23 views

Verifying that an ideal which avoids a certain set is a prime ideal

Let $R$ be a commutative ring with $1 \neq 0$. Assume that $a \in R$ is such that $a^n \neq 0$ for each positive integer $n$ and let $\mathcal S = \{a^n\}_{n \geq 0}$. Prove that there exists an ...
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1answer
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Irreducible algebraic sets with intersecting parts

Let $V = V(F)$ be an irreducible hypersurface in $A^n(k)$. To show: If $W$ is an irreducible algebraic set in $A^n(k)$ with $V \subset W$, then $V = W$. The ideas I got so far: Since $V, W$ are ...
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subring of rational numbers and its ideal

Let $p$ be a prime number. For any $p$ the subring $\mathbb{Q}_p$ of of the field of rational numbers is defined: $\mathbb{Q}_p=\{\frac{a}{b}|a,b\mbox{ are integers, $p$ does not divide $b$}\}$ Let ...
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If $R\to S$ is a ring homorphism with $J$ an ideal of $S$. Show that the preimage of $J$ is an ideal of $R$.

Let $\alpha\colon R\to S$ be a ring homomorphism. Let $J$ be an ideal of $S$, and define the preimage of $J$ by $\alpha^{−1}(J)=\{r\in R\mid \alpha(r)\in J\}$. Show that $\alpha^{-1}(J)$ is an ideal ...
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136 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 ...
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2answers
282 views

$\left<2,x\right>$ is a maximal ideal of $\Bbb Z[x]$

I want to show that $\left<2,x\right>$ is a maximal ideal of $\Bbb Z[x]$. My game plan is to use the 3rd isomorphism theorem to somehow get that $Z[x]/\left<2,x\right>$ isomorphic to ...
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1answer
101 views

How to show that a subset of a Ring is an ideal

My book says that $I$ is an idea of a right $R$ if $I$ is an additive subgroup of $R$ $ra \in I$ and $ar \in I$ for any $r\in R$ and $a \in I$ Both conditions are straightforward checks, but I ...
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47 views

Showing that if commutative R has infinite strictly ascending chains of principal ideals then so does R[x]

First of all, I don't understand why $f_{i+1}\mid f_i$. If $\left< f_i \right>\subset \left< f_{i+1} \right>$, then wouldn't $f_{i}\mid f_{i+1}$? Thank you for your help so far. I have ...
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2answers
457 views

Relatively prime ideals/comaximal ideals

Let $R$ be a commutative ring. a) Prove that if $I$, $J$ are proper ideals of $R$ with $I + J = R$ then for any $a, b \in R$ there exists $x \in R$ such that $x + I = a + I$ and $x + J = b + ...
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1answer
62 views

Showing that an ideal is not principal

I supposed to the contrary that $I$ is principal, so then $2$ and $1+\sqrt{-3}$ must have a common factor. However, since $2$ and $1+\sqrt{-3}$ are irreducible, the only common factors are $1$ and ...
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2answers
141 views

Maximal ideals in $\mathbb Z[i]$

Let $\mathbb Z[i]=\{a+bi \mid a,b∈ℤ\}$ be a subring of $ℂ$. Consider two principal ideals $I=(7)$ and $J=(13)$ in $\mathbb Z[i]$. Is the ideal $I$ maximal? Is the ideal $J$ maximal?
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1answer
67 views

Why is (gcd(f,g)) = (f,g)?

f and g are polynomials of F[X]. I can't see why (f,g) = (gcd(f,g)) ? (the ideal that f and g are the generators, and the ideal that the gcd is the generator). gcd(f,g) = a*f+b*g , for specific a ...
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1answer
99 views

Ring homomorphism, maximal ideals

Here's a question from my worksheet, i solved subquestion (1) but can use help with the other 2...And also would appreciate any comments on my answer for subquestion (1). Let $\psi: R->S$ be a ...