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

6
votes
1answer
250 views

In $K[X,Y]$, is the power of any prime also primary?

I've recently been reading about primary decomposition, and was browsing the questions here. From this, I know that it is not true that every primary ideal is the power of a prime ideal. I'm curious ...
6
votes
1answer
273 views

Is there an example of commutative ring with exactly three prime ideals for which this property holds?

Is there an example of commutative ring with exactly three non zero prime ideals $P_i$ which satisfies the following statement: $P_1P_2=0$ and for an ideal $I\neq 0$ such that $I\neq P_i$ we have ...
6
votes
1answer
87 views

When is k[x,y]/I complete for the (x,y)-adic topology?

Let $k$ be a field. If necessary, add assumptions on $k$ or just take $k=\mathbb{C}$. It is easy to classify the ideals $I \subseteq k[x]$ such that $k[x]/I \to k[[x]]/(I)$ is an isomorphism, namely ...
6
votes
1answer
539 views

Annihilator of quotient module M/IM

Let $A$ be a commutative ring, $I$ an ideal of $A$ and $M$ an module over $A$. Is it true that $\operatorname{Ann}(M/IM) = \operatorname{Ann}(M) + I$? One inclusion is certainly true, but I ...
6
votes
1answer
120 views

$p^2=p\in \bar{I}$, I ideal of Banach algebra $\Rightarrow p\in I$

Let $I\subset A$ be a ideal of a Banach algebra $A$. Assume $p\in \overline I$ and $p^2=p$. Show: $p\in I$. Can someone give me a little hint how to solve this, please?
6
votes
1answer
88 views

P(R) is contained in Nil(R) for noncommutative rings.

How to show that $P(R)$ is contained in $\operatorname{Nil}(R)$ (where $R$ is a noncommutative ring with identity)? Definitions I am using: A nil right ideal is one whose elements are all ...
6
votes
2answers
286 views

(Minimal?) Polynomials using the Nullstellensatz

I'm struggling with an exercise that was asked in class: Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
6
votes
3answers
83 views

Let $R = \mathbb Z[i]$. Show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$.

Let $R = \mathbb Z[i]$, $z = 3+i$ and $I = \langle z \rangle$. I need to show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$ and $10 \mid a$, ...
6
votes
1answer
129 views

Factorization of $5$ in the splitting field of $x^3 + 2$

I wonder if someone could help to clarify the following. Let $\zeta$ be a primitive cube root of unity and $\alpha = \sqrt[3]{2}$. Let $K = \mathbb{Q}(\alpha)$ and $L = K(\zeta)$. Then $L$ is the ...
6
votes
2answers
1k views

An ideal that is radical but not prime.

I'm preparing for an exam and, as part of this preparation, I'm looking for an ideal $I$ in an integral domain $R$ that is radical but not prime. Here is an example I'm fooling around with: ...
6
votes
1answer
448 views

Number of generators of the maximal ideals in polynomial rings over a field

Hi I'm trying to prove the following If $K$ is a field (not necessary algebraically closed) then every maximal ideal of $K[x_{1},\dots,x_{n}]$ is generated by exactly $n$ elements. I know that ...
6
votes
2answers
465 views

Diophantine equation (use class ideal group to solve)

Use ideal class group to find all integer solutions to the equation $$x^3=y^2+200$$ My approach: Observe that $\mathbb{Z}[\sqrt-2]$ is the field of integers in the ring $\mathbb{Q}(\sqrt -2).$ ...
6
votes
2answers
254 views

A (probably) wrong exercise from Morandi's Field and Galois theory

After some efforts I realize that the following exercise is wrong: (rings are unitary throughout the book) Morandi's Field and Galois Theory, Appendix A, exercise 18 (b) Let $A\subseteq B$ ...
6
votes
1answer
202 views

A Gröbner Basis Computation Gone Bad

Here is the problem statement: Consider the polynomial ideal $I = \langle b-r_1-r_2, c-r_1r_2 \rangle \subset \mathbb{Q}[r_1,r_2,b,c].$ Show that $I \cap \mathbb{Q}[b,c] = \langle 0 \rangle$. ...
6
votes
1answer
49 views

Is there an ideal decomposition that counts the number of monomial generators?

Consider the ideal $I\subseteq S[x,y,z]$ where $S$ is some field of characteristic 0 (probably any field will do) and $I=\langle x^9-y^4z^4,y^9-x^5z^4,z^8-x^4y^5,x^6\rangle$. Notice that because the ...
6
votes
0answers
262 views

Showing an ideal is prime in polynomial ring

Let $k=\mathbb{C}$ and let $J$ the ideal $(xw-yz,y^{3}-x^{2}z,z^{3}-yw^{2},y^{2}w-xz^{2})$. I want to see why $J$ is a prime ideal in $k[x,y,z,w]$. I know that $Z(J)$ (the zero set of $J$) is ...
5
votes
5answers
636 views

Idempotents in a local ring

Is it true that a local ring, i.e., a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
5
votes
4answers
764 views

If $I = \langle 2\rangle$, why is $I[x]$ not a maximal ideal of $\mathbb Z[x]$, even though $I$ is a maximal ideal of $\mathbb Z$?

Let $I = \langle 2\rangle$. Prove $I[x]$ is not a maximal ideal of $\mathbb Z[x]$ even though $I$ is a maximal ideal of $\mathbb Z$. My professor mentioned that I should try adding something to ...
5
votes
3answers
568 views

fields are characterized by the property of having exactly 2 ideals [duplicate]

Possible Duplicate: A ring is a field iff the only ideals are $(0)$ and $(1)$ Michael Artin's Algebra in the introduction of maximal fields, there was a sentence stated that fields are ...
5
votes
3answers
1k views

Ideal of the twisted cubic

The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute ...
5
votes
1answer
472 views

Is any quotient of a Euclidean domain by a prime ideal a Euclidean domain?

Let $R$ be a Euclidean domain, i.e., a ring with a norm $N : R \rightarrow \mathbb N$ such that for any $a,b\in R$ with $b\not=0$, we may write $a = qb + r$ for some $q,r \in R$ with $N(r) < ...
5
votes
1answer
713 views

Are distinct prime ideals in a ring always coprime? If not, then when are they?

Essentially as the title suggests - in some commutative ring $K$ (with 0,1), if we have 2 distinct proper prime ideals $\mathfrak{p}_1 \neq \mathfrak{p}_2$, is it necessarily the case (or if not, when ...
5
votes
2answers
170 views

Is there any non-monoid ring which has no maximal ideal?

Is there any non-monoid ring which has no maximal ideal? We know that every commutative ring has at least one maximal ideals -from Advanced Algebra 1 when we are study Modules that makes it as a very ...
5
votes
2answers
757 views

Maximal Ideals of direct products

Maximal Ideals of $R\times S$ are either of the form $A \times S$, where $A$ is maximal in $R$, or of the form $R\times B$, where $B$ is maximal in $S$. I started by assuming $U$ is maximal in $R ...
5
votes
2answers
231 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 ...
5
votes
2answers
363 views

What is a projective ideal?

I've been looking for the definition of projective ideal but haven't found anything, all I've seen is the definition of projective module (but I don't know how these are related, if they are ¿?). Does ...
5
votes
4answers
542 views

Maximal ideals in $K[X_1,\dots,X_n]$

Let $K$ be a field, and $a_1,\dots,a_n \in K$. Prove that the ideal $$(X_1-a_1,\dots,X_n-a_n)$$ is maximal in $K[X_1,\dots,X_n]$. I tried proving that the only elements outside the ideal are the ...
5
votes
2answers
96 views

Ring whose all ideals are double-sided is commutative?

I was thinking about the following problem: Suppose R is a ring s.t. every left ideal is also right. Is R commutative? This actually continues the easier question: Suppose G is a group whose ...
5
votes
2answers
523 views

Are the determinantal ideals prime?

I want to prove the determinantal ideals over a field are prime ideals. To be concrete: For simplicity, let $I=(x_{11}x_{22}-x_{12}x_{21},x_{11}x_{23}-x_{13}x_{21},x_{12}x_{23}-x_{13}x_{22})$ be ...
5
votes
4answers
204 views

Checking that some ideal is maximal in the multivariate polynomial ring

Let be $k$ a field and $k[x_1,x_2,...,x_n]$ its polynomial ring in $n$ variables. Let be $I$ the ideal generated by $x_1-c_1,x_2-c_2,...,x_n-c_n$, where $c_1,...,c_n$ are elements of $k$. I want to ...
5
votes
3answers
1k views

Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
5
votes
2answers
100 views

Where do I use the fact that $F$ is algebraically closed in this proof?

I have to do the following. Let $F$ be an algebraically closed field. $I\in F[X_1,...,X_n]$ an ideal. Denote by $S(I)$ the subset in $F^n$ consisting of all $n$-tuples $(a_1,...,a_n)\in F^n$ such that ...
5
votes
1answer
409 views

Does every Noetherian ring contain at least one maximal ideal?

I want to prove that a noetherian ring $R \neq \{0\}$ contains at least one maximal ideal. My idea is to consider $\langle 0 \rangle$ and $\langle 1 \rangle$: If there is no ideal $I$ with ...
5
votes
1answer
367 views

Is every proper nontrivial ideal in a Noetherian ring not flat?

I guess my general question is exactly what's in the title, but let me explain why I'm asking and how I came to it. Consider the ideal $I=\langle x,y \rangle \subset k[x,y]$ for a field $k$. Just to ...
5
votes
4answers
160 views

Why is $(2, 1+\sqrt{-5})$ not principal?

Why is $(2, 1+\sqrt{-5})$ not principal in $\mathbb{Z}[\sqrt{-5}]$? Say $(2,1+\sqrt{-5})=(\alpha)$, then since $2\in(2,1+\sqrt{-5})$ we have $2\in (\alpha)$, so $\alpha\mid2$ in $\mathbb ...
5
votes
1answer
78 views

Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$

Let $w\in\mathbb C$ be such that $w^3=1$ and $w\neq1$. Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$, and describe $\mathbb Z[w]/(2)$. What I wanted to do is to show that $\mathbb Z[w]$ is a ...
5
votes
2answers
297 views

Vandermonde matrices over a commutative ring.

Suppose that $R$ is a commutative ring with identity. I am trying to prove that the two following statements are equivalent. The ideal generated by all determinants of $n\times n$ Vandermonde ...
5
votes
1answer
340 views

Is there a distributive law for ideals?

I'm curious if there is some sort of distributive law for ideals. If $I,J,K$ are ideals in an arbitrary ring, does $I(J+K)=IJ+IK$? The containment "$\subset$" is pretty clear I think. But the ...
5
votes
2answers
74 views

Uniqueness of prime ideals of $\mathbb F_p[x]/(x^2)$

What are the prime ideals of $\mathbb F_p[x]/(x^2)$? I have been told that the only one is $(x)$, but I would like a proof of this. I want to say that a prime ideal of $\mathbb F_p[x]/(x^2)$ ...
5
votes
2answers
64 views

Definition of primary ideal question

A primary ideal (in a commutative ring with unity) is an ideal $J$ for which if $ab\in J$, then either $a\in J$ or $b^n\in J$ for some integer $n\geq 1$. So it also implies (due to commutativity) that ...
5
votes
1answer
110 views

Minimal Ideal of a Commutative Ring with Unity

Can anyone help me prove this? This one is from Malik's Fundamentals of Abstract Algebra. An ideal $I$ of a ring $R$ is called a minimal ideal if $I≠{0}$ and there does not exist any ideal $J$ of R ...
5
votes
1answer
499 views

Norm of ideals in quadratic number fields

I do not really understand how to factor ideals in a quadratic field $K = \mathbb{Q}(\sqrt{d})$, mainly because I have some trouble computing the norm of ideals. I think I understand what is going on ...
5
votes
1answer
1k views

$I+J=R$, where $R$ is a commutative rng, prove that $IJ=I\cap J$.

So I basically have to prove what is on the title. Given $R$ a commutative rng (a ring that might not contain a $1$), with the property that $I+J=R$, (where $I$ and $J$ are ideals) we have to prove ...
5
votes
2answers
267 views

Characterization of primary ideals in a principal ideal domain

On the commutative algebra wiki, a table of properties lists that "for a PID, the primary ideals coincide with the powers of prime ideals." I played around with it, couldn't produce a proof, ...
5
votes
1answer
310 views

Localization at a Maximal Ideal

While studying, I came across this question: If $A$ is a ring in which $x^n=x$ for all $x\in A$ (where $n$ is an integer greater than $1$ and may depend on $x$), show all prime ideals are maximal. ...
5
votes
2answers
141 views

In an ideal, pairwise non-coprime implies globally non-coprime?

Let $R$ be a polynomial ring $R=k[X_1,X_2, \ldots ,X_n]$. Let $I$ be an ideal of $R$ such that any two elements of $I$ have a non-constant gcd. Does it follow that there is a non-constant $D$ dividing ...
5
votes
2answers
156 views

Computing the radical of an ideal

What is the best way to compute $\sqrt{(X^2-YZ,X(1-Z))}$ ? This is after using Nullstellensatz by the way as I thought it would be easier to compute a radical than finding the vanishing ideal.
5
votes
1answer
97 views

Maximal Ideals of $\mathbb{R}^{\infty}$

In the ring $\mathbb{R}^{\infty}$ (with the standard operations of component-wise addition and multiplication), what are the maximal ideals? It was quite simple to determine that the ideals with a ...
5
votes
1answer
161 views

Endomorphism of a local $k$-algebra inducing an automorphism modulo $m^2$ is an automorphism

The following is exercise 4.1 of Hartshorne's Deformation Theory, used in the proof given there of the sufficiency of the infinitesimal lifting criterion of smoothness: Let $(A,m)$ be a local ...
5
votes
2answers
78 views

Ideal for two polynomials in three variables

Consider the set $B=\{(t^2,t^3,t^4)\mid t\in \mathbb{C}\}$. It is a subvariety of $\mathbb{C}^3$, because it is equal to $V(y^2-x^3,z-x^2)$. How can we find the ideal $I(B)$? I think it is $I(\langle ...