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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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 ...
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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, ...
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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 ...
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What ideal is this?

Let $k$ be a field and $R = k[X]$ all polys over $k$ in $X$. Choose $p \in R$ and define $I_p = \{ f \in R : f\circ p(X) \in I \}$, where $I$ is some ideal in $R$. Then $I_p$ is an additive ...
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Commutative rings whose non-trivial ideals are maximal

It is well known that a local ring is a ring containing only one maximal ideal. I was wondering if there is a characterization (or any information) of the commutative rings such that all their ...
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Subset of a P-ideal need not be a P-ideal

I was looking for examples showing that subset of a P-ideal is not necessary. I will post below a counterexample I was able to find. (I hope it is correct.) But I'd be glad to see other simple (or ...
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Question about ideals of rings and rings of n*n matrices

A question that's been bothering me: R is a ring with unity. Also consider $M_n(R)$ the matrix ring. If all ideals $J$ of $M_n(R)$ are finitely generated, does every ideal $I$ of $R$ need to be ...
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Prime ideals in certain local ring extensions

$(R,m)\subseteq (S,n)$ is a local extension of rings and $S$ is a finitely generated $R$-module. If $P$ is a prime ideal of $R$ such that $P\subset m^2$ and $P'$ is a prime ideal in $S$ such that ...
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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 ...
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Prime ideals in $C[0,1]$ and ultrafilters

I'm looking for prime ideals in the ring $C[0,1]$ of continuous functions $[0,1]\longrightarrow \mathbb{R}$. I raised that question recently and got good answers but now I'd like to improve a bit the ...
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Length of maximal chain of prime ideals equals transcendence degree of fraction field?

I've been reading some commutative algebra, but have been struggling with this idea for a while. Let $k$ be a field, and let $A=k[x_1,\dots,x_n]$ be a finitely generated integral domain, such that ...
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Ideal contained in the union of two ideals and a prime

Taken from Miles Reid "Undergraduate Commutative Algebra" p.35 ex. 1.12 b) Let $I,J_1,J_2 \subset A$ be ideals of a commutative ring $A$. Let $P$ be a prime ideal, then if $I \subset J_1 \cup J_2 ...
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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 ...
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When is intersection of infinitely many maximal ideals zero?

I've been trying without success to figure out what are the rings $R$ such that whenever $M_n, n \in \omega$ is a countably infinite collection of pairwise distinct maximal ideals then $\bigcap_{n \in ...
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Relation between primary ideal and prime ideal

We know that every prime ideal is primary ideal. But can we say, every primary ideal is a power of prime ideal? if it is not correct a counterexample. Thanks.
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Height unmixed ideal and a non-zero divisor

Let $R$ be a commutative Noetherian ring with unit and $I$ an unmixed ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal? Background: A ...
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Ideal generated by two elements that is not free

I am having problems understanding an example constructed by M. Ojanguren and R. Sridharan showing that over the polynomial ring in two variables over a division ring (which is not a field) there ...
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A prime poset of ideals

Let $A$ be a ring (commutative unital), and $\mathcal I$ be a nonempty family of proper ideals of $A$. I will say that $\mathcal I$ has property $\dagger$ if for any $\mathfrak a\in\mathcal I$ and ...
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Quotient of ring of integers

Let $R=\mathcal{O}(K)$ be the ring of the integers of $K=\mathbb{Q}[\zeta_8]$, where $\zeta_8=e^{2\pi i/8}=\sqrt{2}/2(1+i)$ is a primitive eighth root of unity in $\mathbb{C}$. It can be shown that ...
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Idempotent and Hermitian vectors in Group Algebra

Let $C$ be the field of complex number and $G$ a finite group, then define $C[G]$ be a vector space over $C$, with elemnts of $G$ as the basis. Then any element in $C[G]$ can be written as $\sum_{g ...
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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 $I \cap \mathbb{Q}[b,c] = \langle 0 \rangle$. ...
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Is there a ring with the lattice of ideals isomorphic to $(\omega+1)^{\operatorname{op}}?$

In this question, I gave an example of a ring whose lattice of two-sided ideals is order-isomorphic to $\omega+1$. I've been playing a bit with trying to find rings with a given lattice of ideals ...
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Three maximal ideals lying over $3\mathbb{Z}$?

A few weeks ago I asked a question about finding the number of maximal ideals lying above $3\mathbb{Z}$ in $B$, where $B$ is the integral closure of $\mathbb{Z}$ in a splitting extension ...
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$(x,y)$-primary ideals

I want to find all ideals $I$ in $\mathbf{C}[x,y]$ with $\sqrt{I}=(x,y)$ and $\dim_{\mathbf{C}}\mathbf{C}[x,y]/I=2$. I have no clue how to about it, I mean I can write down some examples, ...
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Find a maximal ideal $I$ in the ring $\mathbb{Z}[i]$ such that $\mathbb{Z}[i]/I$ is isomorphic to $\mathbb{Z}/521\mathbb{Z}$.

I know $\mathbb{Z}[i]$, the Gaussian integers, is a PID. So $I$ is generated by a single element. At first I thought $I=(521)$, but $521$ can be reduced to $11^2 + 20^2$. Would $I=(11 + 20i)$ or ...
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Irreducible homogeneous ideals

I have the following question: Let $I$ be a homogeneous ideal. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideals? So, is it ...
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Counterexamples to the avoidance lemma for arbitrary ideals

Let $A$ be a commutative ring with $1$. Let $I$ and $J_k$, $k=1,\dots,n$ be ideals of $A$ with $I\subseteq \cup _{k=1}^n J_k$. Then I have obtained the following: (1) If $J_k$, $k=1,\dots,n$, are ...
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Non-cyclic unit groups

Is there any way to motivate why certain factor rings of $\mathbb{Z}, \mathbb{Z}[i]$, etc., to a prime power have non-cyclic unit groups? For example, the only such non-cyclic unit groups of factor ...
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Why do $f$ and $f'$ generate all of $K[X]$?

I have been studying Marcus' Number Fields. I am stuck on a remark in Appendix 2, page 258. He says: A monic irreducible polynomial $f$ of degree n over $K$ (a subfield of $\mathbb{C}$) splits into n ...
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Find an ideal in $K[x,y]$ that is maximal but not principal.

Let $K$ be a field. Find an ideal of $K[x,y]$ that is maximal but not principal. Prove your claims.(Here we are working in a commutative ring with $1\neq 0.$) My idea: Choose $K=\mathbb{Q}.$ ...
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Proof of the uniqueness of maximal ideal

Let $R$ be a commutative ring with $1$. Let $M$ be a maximal ideal of $R$ such that $M^2 = 0$. Prove that $M$ is the only maximal ideal of $R$.
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The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not principal

The ideal $I= \langle x,y \rangle\subset k[x,y]$ is not a principal ideal. I don't know how to consider it. Any suggestions?
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Commutative ring with an ideal that contains all the nonunits

Is there an example of a commutative ring with an ideal that contains all the non-units? I was trying to think of some subring of $\mathbb Q$, but I couldn't get it to work.
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Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,…,x_6]$ a radical ideal? Is it a prime ideal?

Is the ideal $I=(x_1 x_5 - x_2 x_4 , x_1 x_6 - x_3 x_4)$ of $k[x_1,...,x_6]$ a radical ideal? Is it a prime ideal? thanks
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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 ...
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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 ...
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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 ...
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Prove : If $I = (p(x))$ is a prime ideal in $F[x]$ then $p(x)$ is irreducible.

I have to show : If $I = (p(x))$ is a prime ideal in $F[x]$, where F is a field, then $p(x)$ is irreducible. In the book I use, there is the proof of the converse which uses Euclid's Lemma. I ...
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Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$

This problem is from a practice exam I was working on. What is the cardinality of the quotient $\mathbb{Z}[x]/(x^2-3,2x+4)$ ? Thoughts. If I find a ring that is easier to handle then this then I ...
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Is the number of prime ideals of a zero-dimensional ring stable under base change?

Let $A$ be a zero-dimensional ring of finite type over a field $k$ and let $X= \textrm{Spec} \ A$ be its spectrum. Note that $X$ is a finite set. Suppose that $k\subset K$ is a finite field extension ...
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Ideals of $\mathscr{I}(V) = \text{Hom}(V,V)$

I have been trying this notoriously difficult problem for quite some time but i haven't made any progress. Let $\mathscr{I}(V)$ denote the set of all homomorphisms $f : V \to V$. That is ...
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$R/I$ when $R$ is the ring of real continuous functions

If $R$ is the ring of all real continuous functions on $[0,1]$, I am trying to find $R/I$ where $$I=\{f\in{R}|f(.5)=0\}$$ Showing $I$ is an ideal is not a problem since we're defining addition and ...
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Maximal ideals in $\mathbb Z[i]$

Let $\mathbb Z[i]=\{a+bi \mid a,b∈ℤ\}$ be a subring of $ℂ$. Consider two principal ideals $I=(7)$ and $J=(13)$ in $\mathbb Z[i]$. Is the ideal $I$ maximal? Is the ideal $J$ maximal?
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Definition of principal ideal

This is a pretty basic question about principal ideals - on page 197 of Katznelson's A (Terse) Introduction to Linear Algebra, it says: Assume that $\mathcal{R}$ has an identity element. For $g\in ...
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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 ...
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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 ...
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For an ideal $I$ of $K[x]$, $K[x]/I$ is finitely generated iff $I$ is nonnull

In a book on rational series, a blunt statement is made to the effect that: For $K$ a field, $I$ an ideal of $K[x]$, $K[x]/I$ is finitely generated iff $I$ is nonnull. The statement elaborates ...
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Is this using the first isomorphism theorem for rings?

Let $F$ be a field and $f(x) = x - 1$ and $g(x) = x^2 - 1$. 1) Show that $F[x]/(f(x)) \cong F$ 2) Is ideal $(g(x))$ maximal? Explain your answer. ** I have a feeling that this uses the first ...
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Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?

This is in follow-up to this question. Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the ...
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Product of a principal proper ideal by itself

Let $P$ be a principal proper ideal in an integral domain. Is it $P^2 \subset P$ in general? If yes, how to prove it? For example, if you look at the ideal $(3)=3\mathbb{Z}$ in $\mathbb{Z}$, it ...