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, ...

learn more… | top users | synonyms

1
vote
2answers
93 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 ...
3
votes
1answer
30 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 ...
2
votes
3answers
27 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 $PR_P$ ...
-5
votes
2answers
74 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 $3.2$ ...
0
votes
3answers
24 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 ...
3
votes
0answers
33 views

$R$ 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 ?
2
votes
2answers
43 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 $\langle q(x) \rangle$ ...
-1
votes
0answers
8 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}...
7
votes
1answer
95 views

Show that the ring of integers $A$ of the cubic field $\mathbb Q[x]$ with $x^3=2$ is principal.

Show that the ring of integers $A$ of the cubic field $K=\mathbb Q[x]$ with $x^3=2$ is principal. The hint given in the book is to majorize the discriminant of $A$ by $D(1,x,x^2)$ and then use the ...
0
votes
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 ...
0
votes
2answers
25 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 ...
1
vote
1answer
79 views

Does reduceness of $K[t_1,\dots,t_n]/I$ imply radicality of $I$?

Let $I$ be an ideal in $K[t_1,\dots,t_n]$. Is it true that if the quotient $K[t_1,\dots,t_n]/I$ is reduced then $I$ is radical? We say that a ring $R$ is reduced if $x^2 = 0$ implies $x=0$ for all $x ...
3
votes
2answers
82 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{...
1
vote
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: ...
0
votes
2answers
208 views

Homogeneous ideals are contained in homogeneous prime ideals

Let $I$ be a homogeneous ideal of a graded ring $S$, $I\ne S$. I want to show that there exists a homogeneous prime ideal which contains $I$. I proved the following: Let $T$ be the set of all ...
0
votes
2answers
79 views

How to decompose that ideal?

We have $$I=\left(x^2+2y^2-3,y(x-y),y(y+1)(y-1)\right)\subset\mathbb{C}[x,y]$$ and I would like to decompose it as intersection of simpler ideals. How could I proceed? For example, in this ...
1
vote
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 ...
7
votes
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 ...
0
votes
1answer
84 views

Maximal ideals of the ring $R=C(\mathbb R)$ of continuous functions

Let $R=C(\mathbb R)$ be the ring of continuous functions $f:\mathbb R\to\mathbb C$ where the addition and the product is pointwise defined. Let $$\mathbb m_a=\{f\in R\ |\ f(a)=0\}$$ be a maximal ...
0
votes
0answers
29 views

Relation between minimal primes of a Noetherian graded ring and its subring

Let $A=⊕A_i$ be a Noetherian graded ring. Is there any relation between minimal primes of $A$ and minimal primes of $A_0$ (its $0$-th component)? In fact, my motivation is tight closure theory. I am ...
4
votes
0answers
48 views

Left ideals of $M_2(K)$ with $K$ a field

Is it true that the only proper left ideals of $M_2(K)$, the ring of the matrices whose coefficients are in a field $K$, are $$ \left\{\begin{pmatrix}ah & ak \\ bh & bk\end{pmatrix}: a,b \in K\...
-1
votes
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 ...
1
vote
0answers
19 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 ...
0
votes
1answer
27 views

A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. [closed]

I expect that the following result is true, but i can't prove it. A homogeneous principal prime ideal in $K[x_1,\dots,x_n]$ is generated by a homogeneous element. I need some help to prove this....
3
votes
1answer
25 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{...
2
votes
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
votes
1answer
21 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 $R$ ...
1
vote
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 ...
0
votes
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 ...
0
votes
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
votes
0answers
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
votes
0answers
32 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 ...
0
votes
0answers
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:...
0
votes
1answer
89 views

A question about ideals of rings

In ring $\mathbb{Z}/2\mathbb{Z}$, which polynomial is in the ideal generated by $1+x^2$ and $1+x^3$ $\mathrm{A}. 1+ x^4 \\ \mathrm{B}. x^5+x+1 \\ \mathrm{C}. 1+x^6$ This type of questions confused ...
3
votes
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 ...
1
vote
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 =...
0
votes
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> $ ...
0
votes
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 ...
0
votes
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 ...
16
votes
2answers
4k views

One-to-one correspondence of ideals in the quotient also extends to prime ideals?

I'm beginning to learn some Grothendieck's algebraic geometry and I have a doubt about a property of commutative algebra. For a comm. ring $A$ and an ideal $I$ of $A$, does the one-to-one ...
1
vote
2answers
62 views

Let $R$ be a ring, $S$ a subring and $I$ an ideal. If $R$ is Noetherian, are then $S$ and $R/I$ also Noetherian?

Let $R$ be a ring, $S$ a subring and $I$ an ideal. If $R$ is Noetherian are then $S$ and $R/I$ also Noetherian? I have done the following: $R$ is Noetherian iff each increasing sequence of ideal $...
4
votes
2answers
862 views

Prove the intersection of localizations at maximal ideals is $A$ [closed]

Let $A$ be an integral domain with field of fractions $K$. How can I prove that $\bigcap\{A_{\mathfrak m}\mid\mathfrak m \subseteq ​​A\text{ is maximal}\} = A$?
1
vote
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 ...
1
vote
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 $S^{-1}r(I) = r(S^{-1}I)$? ...
-2
votes
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 ...
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 ...
0
votes
1answer
34 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 ...
1
vote
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,...
1
vote
1answer
44 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. ...