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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If singular set is finite then the ideal is radical

Let $F\in K[X,Y]$ and if the zero set $V(F,\frac{\partial F} {\partial x},\frac{\partial F} {\partial y})$ is finite then $\sqrt {(F)} = (F)$. I don't see the relation between $\frac{\partial F} ...
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Counterexample: multiplying modules by elements of an ideal vs. taking linear combinations

Let $R$ be a ring (commutative, unital) and $M$ an $R$-module. Let $I \subset R$ be an ideal. We make the following definitions: $$ A := \{ am \ | \ a \in I,\ m \in M \} $$ $$ B := \left\{ ...
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46 views

Factor ring of polynomial

$F[x]$ is a polynomial ring over a certain field $F$. $J$ is an ideal of $F$, $J = (f(x))$. I need to prove that if the polynomial $f(x)$ has a multiple root the factor ring $F[x]/J$ is not a field. ...
3
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237 views

Find maximal ideals of a ring

I'm trying to find all the maximal ideal of the ring $\mathbb{Z}[\sqrt{3}] = \{a+b\sqrt{3} : a,b\in \mathbb{Z}\}$. I have found one, that is $A = \{3a + b\sqrt{3} : a,b \in \mathbb{Z} \} $, and I ...
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Prove that this factor ring is a finite ring without zero divisors [duplicate]

Let $R=\mathbb{Z}[\sqrt{d}]=\{a+b\sqrt{d}\mid a,b,d\in\mathbb{Z}\}$ and let $d$ be square-free. Let $P$ be a non-zero prime ideal in $R$. I need to prove that the factor ring $R/P$ has no zero ...
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80 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
3
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1answer
73 views

$\mathbb Z[i]$ and the ideal $(5)$ [closed]

Consider $\mathbb Z[i]$ the ring of Gaussian integers and its ideal $J=(5)$. Show that $\mathbb Z[i]/J \cong \mathbb Z_5 \oplus \mathbb Z_5$ as rings.
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38 views

Algebra “A/I”: what does it mean?

I have an Algebra $A$ and an Ideal $I \subset A$. What is the algebra $A/I$ ? I've seen it several times but I can't find the definition... Maybe is it the subspace of $A$ that is $A \ominus I$ ?
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1answer
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Showing $\{a+b\sqrt{2} \in R$ | $a$ is divisible by $2\}$ is an ideal.

This was a problem in my math book and I was wondering what the proof looks like: Let $R = \{a + b\sqrt{2} | a,b\in \mathbb{Z}\}$. Show that $I = \{a+b\sqrt{2}\in R | a$ is divisible by $2\}$ is an ...
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63 views

Describe units and maximal ideals in these two PIDs

If $p$ is a fixed prime integer, let $R$ be the set of all rational numbers that can be written in the form $(a)$ $\frac{a}{b}$ with $b$ not divisible by $p$. $(b)$ $\frac{a}{b}$ with $b=p^k$ for a ...
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649 views

Prove that every nonzero prime ideal is maximal in $\mathbb{Z}[\sqrt{d}]$

$d \in \mathbb{Z}$ is a square-free integer ($d \ne 1$, and $d$ has no factors of the form $c^2$ except $c = \pm 1$), and let $R=\mathbb{Z}[\sqrt{d}]= \{ a+b\sqrt{d} \mid a,b \in \mathbb{Z} \}$. ...
2
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3answers
119 views

Products of ideals is an ideal and comaximal ideals

Let $I$ and $J$ be two ideals in a ring $R$. Prove that $IJ$ is an ideal. Prove that if $R$ is a commutative ring with two ideals satisfying $I+J=R$ then $IJ=I\cap J$. I could prove that $IJ$ has ...
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2answers
47 views

Ideals of $ \mathbb{Z}/n\mathbb{Z} $

Let $n \in \mathbb{N}$ and suppose there is some positive integer $k \neq 1$ such that $k^2\mid n$. Can I prove that there exists two distinct ideals $I $ and $J$ of $ \mathbb{Z}/n\mathbb{Z} $ such ...
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1answer
55 views

Is there an Artinian ring with exactly two prime ideals which their product is non-zero?

Is there an Artinian ring with exactly two prime ideals which their product is non-zero? Clearly these prime ideals could not be zero on the other hand the summation of them is equal to R.
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For an ideal $J$ is it true that $J \circ J = J$?

My question is pretty straightforward. If for a ring $\left({R, +, \circ}\right)$ we have an ideal $J$, is it true that $J \circ J = J$? In other words $\forall j \in J \quad \exists j_1,j_2 ...
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135 views

Describe units and maximal ideals in this ring

If $p$ is a fixed prime integer, let $R$ be the set of all rational numbers that can be written in a form $\frac{a}{b}$ with $b$ not divisible by $p$. I need to describe all the units in $R$ and ...
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1answer
110 views

No nonzero proper ideals of $K$-algebra $A$ implies the ring $A$ has no nonzero proper ideals

This is from Seth Warner's Classical Modern Algebra. The problem is: If $A$ is a nontrivial $K$-algebra possessing no nonzero proper ideals, then there are no nonzero proper ideals of the ring ...
2
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1answer
58 views

$I$ finitely generated ideal with $I=I^2$, then $I$ is a direct summand of $R$

Let $R$ be a commutative ring, and let $I$ be a finitely generated ideal in $R$ with $I=I^2$. Show that $I$ is a direct summand of $R$. I managed to understand that $(1-z)I=0$ for some $z \in ...
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2answers
85 views

Proving idempotent ideals are solutions to $x^2 = x$

Let $I$ be an ideal in ring $R$. Prove that every element in $R/I$ is a solution of $x^2=x$ if and only if for every $a$ in $R$, $a^2-a$ is in $I$. Let $a \in R/I$. Suppose $a$ is a solution to ...
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What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
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1answer
51 views

Short question on notation of ideals in a quotient ring.

Let $R = \Bbb C[x]/J$. Then the ideals of $R$ are in 1-1-correspondence with the ideals of $\Bbb C[x]$ that contain $J$. Since $\Bbb C[x]$ is a PID then every ideal is principal. Let $I=\langle ...
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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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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
174 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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2answers
61 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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1answer
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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
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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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1answer
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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
395 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
53 views

Are two different prime ideals relatively prime?

Are two different prime ideals relatively prime? Thanks in advance!
0
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1answer
66 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.
3
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1answer
66 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. ...
3
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1answer
80 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 ...
3
votes
1answer
224 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 ...
2
votes
2answers
264 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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2answers
67 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 ...
3
votes
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
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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
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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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0answers
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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 ...
0
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1answer
89 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 ...
1
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1answer
66 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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1answer
63 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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votes
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239 views

$\mathbb Z\times\mathbb Z$ 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 ...
3
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1answer
113 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 ...
2
votes
1answer
39 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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votes
2answers
104 views

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 $R$, $\sqrt{P^{n}}=P\ \forall n\in\mathbb{N}$?
2
votes
3answers
110 views

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 ...
5
votes
2answers
179 views

$R$ 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 ...
0
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61 views

Ideals in cubic fields

I've been studying number fields, and the ideals of their integer rings, and I have a question. First, I know the following in the quadratic case. If a $\mathbb{Z}$-basis for the integer ring is ...