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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$I=mI$, when $I$ is not finitely generated.

Let $(R,m)$ be a commutative local ring with unit. Suppose $I$ is an ideal (not finitely generated). If $I=mI$, what can we say about $I$? If $I$ were finitely generated, then Nakayama's lemma would ...
4
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42 views

problems about prime ideals

Let $A=\mathbb{Z}[X,Y]/(Y^2-6X^2), B=\mathbb{Z}[X,T]/(T^2-6)$ where $X,Y,T $ are variables and let $x,y$ be the cosets of $X,Y$ in $A$ whilst $x',t$ be the cosets of $X,T$ in $B$. Consider the ideals ...
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30 views

A non domain ring with every non-zero annihilating ideal a prime ideal has a particular form.

A non-domain ring in which every non-zero annihilating ideal is a prime ideal, is of the form $F_1 \bigoplus F_2$, $F_1$, $F_2$ are fields or has only one non-zero proper ideal. Note: Here, an ideal ...
4
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1answer
45 views

Units in quotient rings

Let $I$ and $J$ be ideals of a ring $R=\mathbb{K}[X_1, \dots, X_m]/K$, quotient of a polynomial ring over a field $\mathbb{K}$. Consider the map $$\begin{aligned} (R/I)^*\oplus (R/J)^*&\...
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1answer
28 views

$R$ be an infinite commutative ring such that $R/I$ has only finitely many ideals for every non-zero ideal $I$ , what can we say about $R$?

It is known that if $R$ is an infinite commutative ring such that for every non-zero ideal $I$ , $R/I$ is finite then $R$ is a Noetheian domain . It is also known that if $R$ is a PID then for every ...
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11 views

relationship between independence of multivariate polynomials, generating sets of polynomial ideals

I am studying something that touches on Groebner algorithms at the moment and It seems like i am missing something obvious about the relationship between three definitions that feel like they should ...
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1answer
33 views

Krull dimension of a quotient by a system of parameters

Let $(A,\frak m)$ be a local Noetherian ring and let $x_1,\dots,x_d$ be a system of parameters, i.e. ${\frak m}=(x_1,\dots,x_d)$. Then $$\dim A/(x_1,\dots,x_i)=d-i$$ for each $i=1,\dots,d$. I know ...
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1answer
21 views

Is collection of all functions I-convergent to a point form a ring?

$S$ be a set. $I$ is an ideal of $S.$ $X$ is a topological space. A function $$f: S\rightarrow X$$ is said to be $I$-convergent to a point $x\in X$ if $$f^{-1}(U)=\{ s\in S; f(s)\in U\}\in \mathscr F(...
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How is a sequence not converging usually but $I_{\tau}$ converging in this given paper.

I am reading the paper Pratulananda Das and Ekrem Savas: On I-convergence of nets in locally solid Riesz spaces, Filomat 27:1 (2013), 89–94, DOI: 10.2298/FIL1301089D. I am stuck at example $3.2$ ...
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3answers
37 views

$(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$

Let $(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$. I was thinking to use Nakayama lemma as: $R_P$ is local with $PR_P$ ...
4
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1answer
43 views

$R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

Let $R$ be a Noetherian domain, $t\in R$ be a non-zero, non-unit element, then is it true that $$\bigcap_{n \ge 1} t^nR=\{0\} \text{?} $$ It almost feels like the nilradical (which is zero for any ...
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3answers
26 views

Minimal ideal in a ring which is generated by an idempotent element.

Let $R$ be a commutative ring with unity and $M$ be a minimal ideal of $R$ such that $M = Re$ where $e$ is an idempotent element in $R$. Then $R = Re \oplus R(1-e) $ I am not able to see, in order ...
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2answers
83 views
+50

$R$ be an integral domain , $x \in R$ , $I$ an ideal such that $I+\langle x \rangle , (I:x)$ are principal ideals , then is $I$ a principal ideal?

Let $R$ be an integral domain , $x \in R$ , $I$ be an ideal such that $I+\langle x \rangle $ and $(I:x):=\{r \in R : rx \in I\}$ both are principal ideals , then is $I$ also a principal ideal ?
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Polynomial in Compact Polytope: Algebraic Description for the Compact Polytope?

Consider a polynomial $f\in K[x_1,\ldots, x_n]$ where $K=\mathbb R$. For example $$f[x_1,x_2,x_3]=x_1 x_2+x_3$$ \begin{eqnarray*} x_{1} & \in & [0.2,0.5]\\ x_{2} & \in & [0,1]\\ x_{3}...
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2answers
46 views

What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
0
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2answers
28 views

Uniqueness of generator of principal ideal in K[x_1,x_2,…,x_n]

In $K[x]$ (where $K$ is a field), I know that every ideal can be written as $(f)$ for some $f \in K[x]$. Furthermore, $f$ is unique up to multiplication by a nonzero constant in $K$. Is there a ...
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2answers
51 views

Jacobson radical of $R=\{\frac{a}{b}:a,b \in \Bbb Z,b \neq0\text{ and }p\nmid b\}$.

I want to find the Jacobson radical, $J(R)$, of $R=\{\frac{a}{b}:a,b \in \Bbb Z,b \neq0\text{ and }p\nmid b\}$. Here my idea: One could use the definition $J(R)$ $=$ {$x \in R;\,\,\forall y \in R: ...
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1answer
59 views

Obtain dimension of multivariate polynomial quotient ring?

Let $\mathbb{C}[z_1,z_2,...,z_n]$ be the ring of multivariate polynomials in complex variables $z_1,z_2,...,z_n$ with complex coefficients. This ring is spanned by the countably infinite basis of ...
3
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2answers
83 views

All ideals of a subring of $\Bbb Q$

Let $p$ be prime and $R=\{\frac{a}{b}:a,b \in \Bbb Z,b \neq0\text{ and }p\nmid b\}$. As an exercise, I have to prove that $R$ is a subring of $\Bbb Q$. My idea: With $a = 1$ and $b = 1$, $\frac{...
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1answer
56 views

If $B$ is an ideal of $A$ then $B[x]$ is an ideal of $A[x]$ - what's wrong with my proof?

This is exercise E.2 from chapter 24 of Pinter's A Book of Abstract Algebra: If $B$ is an ideal of $A$, $B[x]$ is not necessarily an ideal of $A[x]$. Give an example to prove this contention. It ...
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1answer
24 views

Showing that the radical contains the original ideal

Let $I$ be an ideal in a commutative ring $R$ and let $$ J = \{ r \in R \mid \text{$r^n \in I$ for some positive integer $n$}\}. $$ Prove that $J$ is an ideal that contains $I$. I can prove ...
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0answers
21 views

What are minimal paths, generators of graph ideal in a cyclic graph $C_n$?

Minimal cuts are the generators of the cut ideal while the Alexander duality of path ideal generated by the minimal paths is the cut ideal -- more on Graph ideals here. Graph ideals are special case ...
3
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1answer
26 views

Field and ideal notation: double bracks/parens vs single brackets/parens

I'm reading some notes that has the following denotation: the set of formal power-series with coefficients in $\mathbb{F}_p$ is denoted by $\mathbb{F}_p[[t]]$. the fraction field, $\operatorname{...
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1answer
50 views

A doubt about the correspondence theorem.

Let $f$ be a ring homomorphism from $R$ onto $R_1$. Then there is a one one correspondence between the set of all ideals of $R_1$ and the set of all ideals of $R$ that contain the kernel. Now what ...
2
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1answer
22 views

Field is an Artinian module

I am going through theorem 2.14 in Eisenbud's Commutative Algebra. Given a ring $R$ that is Noetherian, all of whose prime ideals are maximal, we want to prove that $R$ is Artinian. Assume that $R$ ...
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2answers
25 views

Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$

Let $\phi: R \to S$ be a surjective homomorphism. Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$ for an ideal $J$ of $R.$ Obviously, $S \cong R/ \ker \phi$(first ...
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2answers
54 views

Intuitive reasons of ring modulo maximal ideal or prime ideal

Are there any intuitive reasons that can help us remember that $R/I$ is a field iff $I$ is a maximal ideal; $R/I$ is an integral domain iff $I$ is a prime ideal? (I can understand the proof, but have ...
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10 views

What are Hilbert Series on Graph Ideals for?

Partially related on Hilbert Series of Monomial ideals but I want to understand the purpose of Hilbert Series on Graph Ideals. Example on the cycle graph $C_4$ with $x_1,x_2,x_3$ and $x_4$ in corners:...
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0answers
33 views

What does Hilbert series of Monomial ideal describe?

I am trying to understand the point of hilbert series of monomial ideals. I am confused because Macaulay has commands for hilbertSeries, hilbertPolynomial and hilbertFunction. What does Hilbert ...
3
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2answers
37 views

$R/(IJ)$ is reduced $\Rightarrow IJ = I \cap J$ for ideals $I,J$ of a commutative ring $R$

This is exercise $4.6$ on page $154$ of the textbook Algebra: Chapter $0$ (authored by P. Aluffi): Let $I,J$ be ideals of a commutative ring $R$. Assume that $R/(IJ)$ is reduced (that is, it has ...
0
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1answer
19 views

Elimination of variables in zero-dimensional ideals

Suppose $f,g \in k[x,y]$ and $I:=\left<f,g\right>$ is an ideal. My question is if the second implication is true. $I$ is zero dimensional $\implies$ $I \cap k[x] \neq \left<0\right> $ ...
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15 views

Demonstrations on the Stanley-Reisner Ideal of Simplical Complex of Graph

The simplicial complex of graph is defined here and I want to understand its Stanley-Reisner ideal where I cannot understand the point "such that there is no face of $\Gamma$ with vertices $x_{...
0
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1answer
26 views

Demonstrations on the Simplicial complex of Graph

where I cannot understand $F\in\Gamma\land G\subseteq F\Rightarrow G\in\Gamma$. I would like to see an example about the simplicial complex of a graph such as a cycle graph $C_3$. What are ...
0
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2answers
41 views

$K[X,Y]$ is a PID and a primary ideal in it is not power of a maximal ideal?

I wonder if $M$ is a maximal ideal, $Q$ is an ideal of $R$ and $\sqrt{Q}=M$ then $Q$ is an $M$-primary ideal. The converse is not true (I know that). We also have that if $R$ is PID which is not ...
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1answer
88 views

$C^*(X)\cap C^P_{\infty}(X)$ is an $e-$ideal of $C^*(X)$

Definitions: $X$ is a completely regular Hausdorff topological space ; $C(X)$ is the set of all continuous function from $X$ to $\mathbb R$ and $C^*(X)$ is the set of all real valued bounded ...
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4answers
45 views

For an algebraically closed field $k$, an ideal $I$ of $k[x]$ is maximal if and only if $I = (x-c)$

This is an exercise $4.21$ on a page $155$ from a textbook "Algebra: Chapter $0$" by P.Aluffi. Let $k$ be an algebraically cloased field, and let $I \subseteq k[x]$ be an ideal. Prove that $I$ is ...
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2answers
116 views

Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
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1answer
40 views

Generator of intersection of ideals in a PID via adjunction?

In a PID we have the formulas $ \left\langle f\right\rangle + \left\langle g \right\rangle = \left\langle \gcd(f,g) \right\rangle $ and $ \left\langle f\right\rangle \cap \left\langle g \right\rangle =...
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1answer
45 views

$(x_1, …, x_k)$ is prime in $R[x_1, …, x_n]$ if $R$ is an integral domain

Let $R$ be an integral domain. I need to prove that $\forall k = 1, ..., n \ \ \ (x_1, ..., x_k)$ is prime in $R[x_1, ..., x_n]$. I managed to do it for $k = 1$. Let $f, g \in R[x_1, ..., x_n]$. Then ...
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1answer
48 views

Does localization commute with taking radicals?

Let $A$ be a ring, $S\subset A$ a multiplicative set, and $I\subset A$ an ideal not intersecting $S$. For any ideal $J$, let $r(J)$ denote the radical of $J$. Is $S^{-1}r(I) = r(S^{-1}I)$? ...
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2answers
37 views

$(f_1(x), f_2(x), …, x-a) = (f_1(a), …, f_r(a), x-a)$ with $a \in R$, $R$ is a commutative ring, $f_i(x) \in R[x]$ [closed]

Let $R$ be a commutative ring, $a \in R$, and $\forall i = 1, ...,r \ \ f_i(x) \in R[x]$. Prove the equality of ideals $(f_1(x), ..., f_r(x), x-a ) = (f_1(a), ...f_r(a), x-a)$. That is, $\forall ...
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1answer
35 views

Expressing a hypersurface of a variety as zero locus

It should be obvious from the question that I am not an algebraic geometer, and so I would really appreciate an answer without using schemes or functor. Let $V$ be an (embedded) variety in a complex ...
0
votes
1answer
23 views

The ideal generated by $(x,y) \cdot (x,z) \cdot (y,z)$ in $k[x,y,z]$ for a field $k$

Let $A=k[x,y,z]$, and let $I$ be generated by $(Ax+Ay)(Ax+Az)(Ay+Az)$. I wish to find a set of three generators for $I$. My first approach for this was by expanding out. It seems that by expanding ...
3
votes
1answer
85 views

Build a reduced ring starting from an ordinary one

This may be easier than I think, but still I can't seem to wrap my head around it. I've learnt that if we take a ring $R$ and quotient it for a (two-sided) ideal $I \subset R$ which is radical, the ...
1
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1answer
45 views

Ideals of Unique Factorization Domain

Let R be a commutative ring with unity such that R[x] is UFD. The ideal (x) of R[x] is denoted by I. Then pick the correct statements from below: 1. I is prime. 2. If I is maximal then R[x] is a PID. ...
0
votes
1answer
30 views

Let $R$ be a commutative ring. For $a \in R$ consider the set $(a) = \{r*a | r\in R\}$. Show that $(a) = R$ if a is a unit of $R$ [duplicate]

I tried some values and I think I got the idea. R is the set of values used in the ring. If I use $\mathbb{Z}$, the units are $\{-1,1\}$. If I take 1 for example, I could use it to get every value in $...
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1answer
51 views

Decomposition of a monomial ideal

I have to find a primary decomposition of the following ideal and I proceeded in this way: $$(x^2z,x^2y^3,xt^2)=(x)\cap(t^2,x^2z,x^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,z^2y^3)=(x)\cap(t^2,x^2)\cap(t^2,z,...
3
votes
2answers
130 views

When a prime ideal in polynomial ring over integers is principal [duplicate]

While dealing with a question about a prime ideal $I\subset\mathbb{Z}[x]$ (with $0$ in $I$ as the only constant polynomial) I was asked to show that there exist $f(x)\in\mathbb{Z}[x]$ such that $I=\...
5
votes
1answer
47 views

Finding nilpotent elements in a quotient ring.

Which are nilpotent elements of $\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)$? I tried to decompose in this way: $$\mathbb{Q}[x]/(x^5-3x^2)\times\mathbb{Z}/(12)\cong\mathbb{Q}[x]/(x^2)\times\mathbb{...
0
votes
1answer
50 views

Principal ideal of $\mathbb{C}[Z,\bar{Z}]$

Let $I$ be an ideal of $\mathbb{C}[Z,\bar{Z}]$. How to prove that $I$ is principal in $\mathbb{C}[Z,\bar{Z}]$ ? It exists some simple criterion to say that an ideal will be principal or not?