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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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 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 ...
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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 ...
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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 ?
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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 ...
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Isomorphism of quotients of powers of maximal ideals

Let $R$ be an integral domain, and $\mathfrak{m}$ a maximal ideal of $R$. Let $R_\mathfrak{m}$ denote the ring localized at $\mathfrak{m}$, and let $\mathfrak{m}_\mathfrak{m} = ...
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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) < ...
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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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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 ...
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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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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 ...
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Is quotient of a ring by a power of a maximal ideal local?

Say I have a commutative ring $R$ with a maximal ideal $m$. Then $m/m^k$ is a maximal ideal in $R/m^k$ for any $k$. Is it the only maximal ideal, i.e. is $R/m^k$ a local ring? This is a well ...
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$R$ 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 ...
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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 ...
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Having trouble with just one line in a proof on why nonzero prime ideals are maximal in a Dedekind domain

http://planetmath.org/?op=getobj&from=objects&name=ProofThatADomainIsDedekindIfItsIdealsAreInvertible In the PlanetMath article above, in the second paragraph of the proof of the first lemma, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\mathbb F_p[x]/(x^2)$ ...
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How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
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Ideal of a ring

I'm trying to describe an ideal of the ring $R=\left\{ \begin{pmatrix}a & b\\ 0 & c \end{pmatrix}:a,b,c \in \mathbb{R}\right\} $ It's easy to prove that $I=\left\{ \begin{pmatrix}0 & a\\ 0 ...
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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 ...
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$I+J=R$, 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 ...
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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. ...
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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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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.
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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 ...
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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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A problem about localization of $\mathbb{Z}/6\mathbb{Z}$ at prime ideal $2\mathbb{Z}/6\mathbb{Z}$

We know that Given a prime ideal $P$ of a commutative ring, there is a one-to-one correspondence between $\lbrace\text{prime ideals }Q\subset P\rbrace$ and $\lbrace\text{prime ideals of } S^{-1}R ...
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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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Every proper ideal contained in a maximal ideal?

This is a true in a commutative ring with $1$, but does it also hold in a noncommutative ring with $1$? The proof in my book is just an application of Zorn's lemma, but the commutativity of the ring ...
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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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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 ...
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Is a square of a prime ideal in a UFD always primary?

More concretely, Let $R$ be a UFD and $\mathfrak{p}$ a prime ideal ideal of $R$. Does it always hold that $\mathfrak{p}^2$ is a primary ideal? I know that it always holds if $\mathfrak{p}$ is a ...
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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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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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If $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $ R $-modules, then $I + J = R$. [duplicate]

If $R$ is a commutative ring with identity and $I$ and $J$ are ideals of $R$ such that $R/I \times R/J$ is isomorphic to $R/(I\cap J)$ as $R$-modules, then $I + J = R$. I know this is the ...
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A relation between the Jacobson radicals of a ring and those of a certain quotient ring

Let $R$ be a ring $J(R)$ the Jacobson radical of $R$ which we define for this problem to be all the maximal left ideals of $R.$ I'm trying to prove the following proposition with only the definition ...
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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 ...