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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Product and intersection of ideals in a polynomial ring

I want to show that in the polynomial ring $K[X,Y,Z,W]$ (where $K$ is a field) the equality $(X,Y)\cap(Z,W)=(XZ,XW,YZ,YW)$ holds. Obviously RHS is contained in LHS. How to show that LHS is contained ...
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0answers
29 views

Determining the kernel of a module homomorphism

Let $p$ be a prime and let $n$ be a positive integer such that $p^n > 2$. Set $R:= \mathbb{Z}_{p^n}$, that is, the residue ring with binary operations of addition and multiplication modulo $p^n$. ...
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1answer
14 views

Equalities of ideals in Q[x,y]

I'm trying to prove something about polinomyal ideals. So I have to use this proposition: Let $I \subset k[x_{1}, \ldots, x_{n}] $ be an ideal, and let $f_{1}, \ldots, f_{s} \in k[x_{1}, \ldots, ...
6
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3answers
49 views

What phenomenon is this? $(2\Bbb{Z} + 1)\cup 3\Bbb{Z} = 2\Bbb{Z} \cup 3\Bbb{Z} + 3$

$(2\Bbb{Z} + 1)\cup 3\Bbb{Z} = 2\Bbb{Z} \cup 3\Bbb{Z} + 3$ Proof: $$ \begin{align*} 2\Bbb{Z} &= \bullet \circ \bullet \circ \bullet \circ \bullet \circ \dots \\ 3\Bbb{Z} &= \bullet \circ ...
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0answers
24 views

There exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$.

$I\trianglelefteq \Bbb F[x]$. I want to prove that there exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$. I guess this means that I am meant to show that we have closure from the ring ...
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2answers
23 views

Difficulty understanding how an element of a quotient ring/field can be represented a certain way…

This is the proposition I'm given, which I don't really understand: Let $p(x)=p_0 + p_1x + ... + p_nx^n$ be an irreducible polynomial over a field $F$, so that $ E = {f[x]}/{\lt p(x)\gt} $ is a ...
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1answer
10 views

Finding generators for products of ideals

If you want to find the generators for the product of ideals, do you simply take all possible products of the generators in the ideals. For example, let $R$ be a ring and let $I = (a,b)$ and $J = ...
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1answer
28 views

Lack of unique factorization of proper ideals in $\mathbb{Z}[\sqrt{-3}]$

I am working on an exercise that asks us to consider the ring $R = \mathbb{Z}[\sqrt{-3}]$ and the ideal $I = (2, 1 + \sqrt{-3})$ in $R$. Part (a) asks to show that $I^2 = (2)I$ but $I \neq (2)$, and ...
6
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2answers
123 views

Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.

Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
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1answer
41 views

Is it true that $I\cap J\subset IJ\subset I+J\subset I\cup J$

Is it true that $I\cap J\subset IJ\subset I+J\subset I\cup J$ If $R$ is a commutative ring and $I,J$ are any ideals of $R$, I don't know how the product is usually defined but I think for $IJ$ is ...
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4answers
45 views

Prove by definition that $(x,2)\subset\mathbb Z[x]$ is a maximal ideal

When the polynomial ring $\mathbb{Z}[x]$ is quotiented by the ideal $(2,x)$ we get a field as $\mathbb{Z}[x]/(x,2)\cong\mathbb{Z}/(2)\cong\mathbb{Z}_{2}$ which is a field. But I ...
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0answers
57 views
+50

A Problem for Nil-Ideals

Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for non-commutative rings I think this may not be true. ...
2
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0answers
48 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
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0answers
102 views

Maps to quotient rings

If $R$ is a ring, $\mathfrak{a}$ is an ideal of $R$ and $S=A[x,y,z,\dots]$ where $A$ is a commutative ring, then is there a ring $S^\prime$ such that there is a bijection: ...
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24 views

Composition of module homomorphisms and their kernels

Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module ...
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2answers
28 views

Inverse images of ideals

I was trying to solve the following exercise: Let $f\colon R\to S$ be a ring epimorphism, $I \subseteq S$ be an ideal, and $J = f^{-1}(I)$. Check that if $I$ is maximal (resp. prime) then $J$ is ...
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3answers
85 views

For a prime integer $p \in \{2, 3, 5, \cdots\}$, is $pR$ a maximal ideal in $R$?

If $R$ is a commutative ring with unit and $p$ is a prime number ($2,3,5,\cdots$), then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?
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2answers
72 views

Proving the Ideal Generated by the Coefficients of $f(X)\cdot g(X)\in R[X]$ is $R$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the ...
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0answers
46 views

Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. [closed]

I took this exercise for a long time but I can't prove it. Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. Anyone could help me?
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0answers
24 views

Module endomorphisms with the same kernel

Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in ...
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1answer
30 views

Modify this formula : $R/I \cong \phi[R]/\phi[I]$

Let $R$ be a ring and $I$ an ideal of $R$, and let $\phi : R\longrightarrow R'$ be a ring homomorphism. Studying by myself, I have a conjecture the following: $$R/I \cong \phi[R]/\phi[I].$$ This ...
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0answers
59 views

How to find prime ideals of $3\Bbb Z$?

Which one of these is prime of $3\Bbb Z$? $42\Bbb Z$,$24\Bbb Z$,$12\Bbb Z$,$9\Bbb Z$ and $33\Bbb Z$ I tried to check their factor groups if they are integral domains. because An ideal I in a ...
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1answer
38 views

Every prime ideal is maximal [duplicate]

Problem: Show that if R is a finite ring, then every prime ideal of R is maximal. My attempt: Let I be a prime ideal of R. Then, by definition of a prime ideal, ab ∈ I implies a ∈ I or b ∈ I for ...
4
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1answer
91 views

finitely generated ideal and number of generators

let $I$ be a finitely generated ideal of ring $R$. Suppose $I/I^2$ as an $R/I$ module is generated by $r$ elements then question is to prove that $I$ is generated by $r+1$ elements.. I have tried ...
0
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1answer
71 views

Maximal ideal of polynomial ring over a subfield

Let $L/K$ be an algebraic extension of fields. Let $B = L[X,Y]$ and $A = K[X,Y]$. Suppose $a$, $b \in L$ and $m = (X-a,Y-b)$ is an ideal of $B$. Show that $m$ and $m \cap A$ are maximal ideals of ...
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1answer
52 views

$M=I\times J $ for some $I,J$ [duplicate]

Let $R,S$ be two rings with identity. Prove that every ideal of $R\times S$ is of the form $I \times J$ where $I$ is an ideal of $R$ and $J$ is an ideal of $S$ . Obviously $I \times J$ is an ideal of ...
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2answers
50 views

Every ideal has a FFR

Let $A$ be a regular local ring. Then every ideal has a finite free resolution. My thoughts: it's easy to prove that every ideal $I$ has a free resolution. In fact $I$ is finite and there is a ...
3
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1answer
42 views

How to show that an ideal is maximal

How do you show that $\langle y^2+2, x-1 \rangle$ is a maximal ideal in $\Bbb Q[x,y]$? I know that if you add another element that is not in this ideal, you should get the whole ring, thus ...
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0answers
27 views

Proof review: Every maximal ideal of ring of continuous functions has the same form

Let $R$ be the ring of real-valued continuous functions on $[0,1]$. If $M$ is a maximal ideal of $R$ prove $\exists \lambda \in [0,1]$ s.t. $M = \{f(x) \in R : f(\lambda) = 0 \}$. (from Herstein ...
3
votes
3answers
34 views

Checking whether $x^2-5$ is prime but not maximal

I want to find an example of a prime ideal that is not maximal. I thought about $x^2-5$. We know that $Z[\sqrt{-5}]\cong Z[x]/(x^2-5)$ is an integral domain, therefore is $x^2-5$ prime. However, I ...
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1answer
19 views

A question on $R$-modules

Let $M$ be a non-trivial irreducible (simple) $R$-module . Let $0 \ne m \in M$ and $A(m_0):=\{x \in R: xm_0=0\}$ , then is $A(m_0)$ a maximal left-ideal of $R$ and as $R$-modules , $M$ and $R/A(m_0)$ ...
2
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1answer
73 views

About Relatively Prime Ideals

I am a physics Master student who has been studying abstract algebra by himself. I have two questions about relatively prime ideals. My first question goes as follows: Let $R$ be a ring, and ...
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1answer
27 views

Prove that $I^2$ is principal.

Consider the ideal $I=(2,\sqrt{-10})$ of $\mathbb{Z}[\sqrt{-10}]$. Prove that $I^2$ is principal. My Try: $I^2=(4,-10,2\sqrt{-10})$. I tried to prove that $I^2=(\sqrt{-10})$. But failed. Is my ...
2
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1answer
31 views

Can MAGMA write Groebner basis elements in terms of the original generators?

Consider the free algebra $F = \mathbb Q(a)\langle x, y, z\rangle$ and the ideal $$I = \langle xy - ayx, yz - zy, xz - zx - y\rangle$$ According to the following code $y^2 \in I$. ...
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0answers
27 views

Confusion regarding a statement in Atiyah-Macdonald

Atiyah-Macdonald says the following: If the ideals $a_i, a_j $ are co prime, then $\Pi a_i= \cap a_i $ What does this even mean? For example, we know that $(2), (3) $ are co prime in the ring ...
2
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2answers
39 views

For $I,J$ ideals $P$ Prime ideal, show that $IJ\subset P \iff I\cap J \subset P$

Question : Prove the following equivalence $IJ\subset P \iff I\cap J \subset P \iff$ $I$ or $J \subset P$ I was able to do this $IJ \subset I$ and $IJ \subset J$ so $IJ \subset P$ $IJ \subset I$ ...
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1answer
30 views

Expressing polynomial as linear combinaion

I found these questions in Adams Introduction to Groebner bases Let $f=x^6-1$ and $g=x^4+2x^3+2x^2-2x-3$. Let $I=\langle f,g\rangle$. Calculate the polynomial that generates $I$ alone. After a ...
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1answer
61 views

Riemann-Roch Theorem and Ideals of a Ring

I found in some Math book a comment stating that the study of Ideals in ring theory à la Dedekind (all kinds of ideals? only one-sided ideals?) could be transferred to other areas (specifically, ...
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1answer
44 views

Some doubts about right ideals of a ring

I would like to know whether the following paragraph regarding right ideals and modules is correct. Any comment or help is welcome: A right ideal of $R$ is just a submodule of the right $R$-module ...
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Converse of Chinese Remainder Theorem

Chinese Remainder Theorem for commutative rings with identity Let $R$ be a commutative ring with identity. If $I, J$ are ideals of $R$ satisfying $I+J=R$, then there is an isomorphism of rings: ...
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1answer
48 views

Proof about the difference between right and left ideals in a ring

I have tried get a version of the proof stating that a left ideals of a ring is not, in general, a right ideal, and viceversa. Is my formulation right? Comments and corrections are welcome. I have ...
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1answer
59 views

Is it true that every prime ideal of height one is principal? [closed]

Is it true that every prime ideal of height one is principal ? Please help
3
votes
2answers
63 views

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$ My brief attempt to try use Bezout theorem at a PID. but unsuccess.. Thanks any help.
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1answer
45 views

Is radical of finitely generated ideal finitely generated?

Let $R$ be non-noetherian commutative ring with identity and $I$ be a finitely generated ideal of $R$; say $I = (a_1, \cdots, a_n)$. Question.1 Is $\sqrt I$ necessarily finitely generated? ...
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2answers
71 views

Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following: Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ ...
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1answer
46 views

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that ...
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votes
1answer
65 views

How does one find the Zariski closure of a set?

I've started to learn algebraic geometry this week (so I do not have much knowledge in the subjet) and, after reading the definition of the Zariski closure $V(I(S))$ of a set $S$, I've tried to do the ...
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0answers
88 views

On a theorem of Akizuki concerning the minimal number of generators of an ideal

I am looking for a theorem of Akizuki I was told by my professor. He said me that Akizuki showed in his paper "Zur Idealtheorie der einartigen Ringbereiche mit dem Teilerkettensatz" (1938) a result ...
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1answer
20 views

Show that in ascending Loewy series, $S^r(R)=R$

Let $R$ be an Artinian ring, $N$ its radical, and $r$ the smallest natural number such that $N^r=0$. Define an ideal $S^n(R)$ of $R$ recursively as follows: $S^1(R)=soc(R)$ Assuming ...
2
votes
1answer
35 views

Factoring ideals in algebraic number rings using Dedekind's theorem

Let $K \subset L=K(\alpha)$ be a number field extension with rings of integers $\mathcal{O}_K$ and $\mathcal{O}_L=\mathcal{O}_K[\alpha]$ respectively. Let $\pi$ be a prime ideal in $O_K$, and let $F = ...