Tagged Questions

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, ...

0answers
33 views

$I=mI$, 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 ...
1answer
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 ...
0answers
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 ...
1answer
45 views

2answers
80 views

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 $3.2$ ...
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 $PR_P$ ...
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 ...
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 ...
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 ?
0answers
16 views

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}...
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 $\langle q(x) \rangle$ ...
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 ...
2answers
51 views

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 ...
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 ...
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 ...
1answer
26 views

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 ...
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 ...
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 ...
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 ...
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 ...
1answer
40 views

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 ...
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 ...
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 ...
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. ...
1answer
30 views