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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Surjective homomorphism of rings. Every ideal of B is an extended ideal of an ideal of A. [on hold]

Let $f: A \rightarrow B$ be a surjective homomorphism of rings. I have to prove that every ideal of $B$ is an extended ideal of an ideal of $A$. Thanks! :)
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100 views

Is product of prime ideals prime?

I'm trying to show that the product of ideals $(x_1, x_3)$ and $(x_2, x_4)$ in $\mathbb C[x_1, x_2, x_3, x_4]$ is a radical ideal, but no other way that I can think of works. So, is the product ...
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$\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A \Rightarrow \mathfrak{a}_{1}^{r_{1}} + \dots + \mathfrak{a}_{n}^{r_{n}} = A$

I have to prove the following : Let $A$ be a commutative ring with unity and let $\mathfrak{a}_{i}$ be ideals in $A$. Assume that $\mathfrak{a}_{1} + \dots + \mathfrak{a}_{n} = A$. Let $r_{i}$ be ...
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Maximal Ideals in $R=\{a+bi:a,b\in \mathbb Z\}$

I've read similar question but please this is not duplicate of Maximal ideals in the ring of Gaussian integers because the answer to it contain PID which I've not yet done etc. $R=\{a+bi:a,b\in ...
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$\overline{\mathbb{Z}}$ is not a Dedekind domain.

I have to prove the following statement : Let $\overline{\mathbb{Z}}$ be the ring of all algebraic integers in (a fixed choice of) $\overline{\mathbb{Q}}$. Then $\overline{\mathbb{Z}}$ is not a ...
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2answers
661 views

Is $(y^2-x)$ a prime ideal in $F[x,y]$?

Let $F$ be a field, and $F[x,y]$ be the ring of polynomials in two variables and we know that $F[x,y]$ is integral domain but not Principal Ideal Domain. We know that $y^2-x$ is irreducible in ...
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Questions related to maximal ideals

In my previous sessional exams, I was asked to prove these two: 1) Find a ring which doesn't have a maximal Ideal. 2) If a ring has unity, then it has a maximal Ideal. About the ...
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61 views

Problem on the number of generators of some ideals in $k[x,y,z]$ [on hold]

I have got stuck with two generator problems: The ideal $(zx,xy,yz)$ can't be generated by $2$ elements. The ideal $(xz-y^2,yz-x^3,z^2-xy)$ can't be generated by $2$ elements. Here the ...
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2answers
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Verifying proof :an Ideal $P$ is prime Ideal if $R/P$ is an integral domain.

I had to write the proof to show that an Ideal $P$ of a commutative ring $R$ is prime Ideal if $R/P$ is an integral domain. let $a,b\in R$ s.t. $ab\in P$ , ...
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31 views

Showing that an ideal is prime

I think that $k[x,y,z]/(z-1, x^2-y)$ can be identified as a subset of $k[x,y]$ with all polynomials whose $x$ terms are only degree one. Therefore I conclude that $k[x,y,z]/(z-1, x^2-y)$ is integral ...
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1answer
29 views

no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$. [duplicate]

$\mathbb Z[i]/\langle 3+i\rangle$ can be represented as :$\{a+3b+\langle 3+i\rangle\big|~~a,b\in \mathbb Z\}$ How shall I find the total no. of elements in $\mathbb Z[i]/\langle 3+i\rangle$.. ...
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1answer
21 views

$A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$

I've a question from gallian which states: Show that $A=\{(3x,y)\mid x,y\in \mathbb Z\}$ is a maximal ideal of $\mathbb Z \oplus \mathbb Z$.Generalize.What happens if $3x$ is replaced by $4x$... ...
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Doubt regarding zero elements in factor ring :$\mathbb Z[i]/\langle3-i\rangle$

I have the factor ring $\mathbb Z[i]/\langle3-i\rangle$ and am asked to find elements zero in this ,they are $0,3-i,i(3-i),(3-i)+i(3-i)$. But I can't understand how do we guarantee these are the only ...
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25 views

Ideal in power series ring

Let $J$ be an ideal in $k[[x_1,...,x_n]]$ such that $(x_{1},...,x_{n})^{2}\subseteq J$, $\{x_{1},...,x_{r}\}\nsubseteq J$ and $\{x_{r+1},...,x_{n}\} \subseteq J$, for some $1\leq r \leq n$. I want to ...
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26 views

An ideal which is not finitely generated

Let $K$ be a field and $A=K[x_1,x_2,x_3,...]$. Prove that the ideal $I:=\langle x_i: i \in \mathbb N\rangle$ is not finitely generated as $A$-module. I have no idea what can I do here, I mean, ...
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1answer
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An Ideal which is Maximal additive subgroup is a Maximal Ideal

How should I prove this: Any Ideal which is a Maximal additive subgroup is also a Maximal Ideal . any idea how to prove it..
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1answer
21 views

Factorization in Dedekind domains

Let $R$ be a commutative, Dedekind (and therefore Noetherian) ring with $1$. Let $I$ be a non-prime ideal of $R$, and let $a,b$ be elements of $R$ such that $a\not\in I,b\not\in I$ but $ab\in I$. Let ...
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A question regarding the number of generators of an ideal [duplicate]

Let $I$ be an ideal in $\mathbb{C}[x_1 ,x_2 ,x_3 ,x_4 ]$ such that $I$ is generated by $x_1 x_3$, $x_2 x_3$, $x_1 x_4$, and $x_2 x_4$. How to show that this I cannot be generated by two ...
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34 views

Ideal and its number of generators [duplicate]

Consider an ideal $I$ in $\mathbb{C}[x_1, x_2, x_3, x_4]$ such that $I$ is generated by $x_1x_3, x_2x_3, x_1x_4,$ and $x_2x_4$. I think this ideal cannot be generated by two elements, but can't ...
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1answer
171 views

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$?

Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=k[X_1,X_2,X_3,X_4]$? Can it be generated with three elements? (Here $k$ is a field.) Thanks for any help.
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Showing that an ideal is maximal

Let $k$ be an algebraically closed field and $f$ be the polynomial $x_1x_2+x_2x_3+x_3x_1$ in $k[x_1, x_2, x_3]$. Here $f$ is irreducible. Then this polynomial ring is not a $PID$, it is only an ...
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10 views

Ideal from ring of fraction

Given $R$ is a commutative ring with $1$ and $D$ is multiplicatively closed containing $1$, I want to show that any ideal of $D^{-1}R$ is of the form $D^{-1}I$, where $I$ is an ideal in $R$. I have ...
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1answer
25 views

If $\cap_{j=1}^{n}I_{j} \subseteq P$ for any ideals $I_1,I_2,..I_n$, then $I_j \subseteq P$ for some $j$

$P$ is a prime ideal if $P$ satisfies the following : If $\bigcap\limits_{j=1}^{n}I_{j} \subseteq P$ for any ideals $I_1,I_2,..I_n$, then $I_j \subseteq P$ for some $j$, where $R$ is a commutative ...
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48 views

Sum and product of comaximal ideals

Let $R$ be a commutative ring with unity. If $R=I_{i}+I_{j}$, for all $i\ne j$, where $I_1,I_2,...,I_n$ are ideals of $R$, I want to show that $$R=I_{n}+I_{1}I_{2}\cdots I_{n-1}.$$ I started off ...
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167 views

Is “P is a left primitive ideal” implies that there is a left maximal ideal…?

By definition, a primitive ideal $P$ exists if there is a simple $R$-module $S$ such that $Ann(S)$=$P$. I saw another statement as follows: "$P$ is a primitive ideal of a ring if there is a left ...
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Ideals of a field

I had the following - apparently straightforward - question on one of my past assignments: Show that a field has no other ideals except $\{0\}$ and the field itself. This was the proof I gave: ...
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1answer
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What exactly does it mean for a maximal ideal to be unique in a principal ideal domain?

I'm currently reading about PIDs and have come across a question involving maximal ideals which at one point reads "Suppose that a Euclidean domain $R$ had a unique maxima ideal $P$". Does this mean ...
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Prove that the intersection of all maximal left ideals of a ring $R$ is a two sided ideal

Prove that the intersection of all maximal left ideals of a ring $R$ is a two sided ideal. What i did:Suppose $B$ be the intersection of all maximal left ideals of the ring $R$. Clearly $B$ is a left ...
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Factorization in noetherian domains

I changed the title (and the body) of this question page, since user26857 provided a nice answer for my original question in a more general setting. Here's what the accepted answer below provides: ...
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How to show that there is a bijective correspondence between two sets of prime ideals

I'm trying to solve this Algebra Problem, and I'm not quite sure, if I'm on the right way. Let $R$ be a commutative ring and $S \subset R$ a multiplicative subset. Show that $p \to pS^{-1}R$ ...
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Primary decomposition of $(XY,(X-Y)Z)$ in $k[X,Y,Z]$

How to find the primary decomposition of $I=(XY,(X-Y)Z)$ in $R=k[X,Y,Z]$? It has minimal primes $(x,y),(y,z),(z,x)$. I tried to calculate $J=S^{-1}I\cap R$, where $S=R-(x,y)$, but it seems ...
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Localizing at maximal ideals and the product

Let $D$ be an integral domain, $M_{i}$, $i = 1,...,r$ be some of its mutually distinct maximal ideals, and $e_{i}$be positive integers for all $i$. Is it true in general that the extension of the ...
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Unique maximal ideal in the ring of fraction

Let $R$ be a commutative ring with 1, and $P$ be a prime ideal in $R$. Let $D = R$ \ $P$. Show that $R_P := D^{-1}R$ has only one maximal ideal. Problem 2b in this link ...
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Find all Gaussian primes in a given range

All $a,b \in \mathbb{Z}$ so that $0\le a \le 4, 1 \le b \le 4$ and $\pi = a+bi$ prime in $\mathbb{Z}[i]$ want to be found. All the possibilities over $\mathbb{Z}[i]$ are : ...
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146 views

What's the motivation of the definition of primary ideals?

$$xy\in\mathfrak q\:\Rightarrow\:\text{either $x\in\mathfrak q$ or $y^n\in\mathfrak q$ for some $n\gt0$}.$$ Primary ideals can be regard as the generalization of prime ideals and radical. But ...
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133 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 ...
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Primality of homogeneous ideal

Let $R$ be the polynomial ring over the finite field $\mathbb{F}_p$ with $n$ variables. Let $I$ be an ideal of $R$ generated by homogeneous polynomials whose coefficients are 1 or -1. Are there any ...
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the sum of two $z^0$-ideals even in $C(X)$ need not to be a $z^0$-ideal

I need to get an example of two $z^0$-ideals while their summation is not? What i know that the sum is a $z^0$-ideal or all of $C(X)$ if and only if $X$ is quasi F-space So i'm searching for an ...
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Irreducible ideals that are not primary.

In my advanced algebra course I've heard that in a noetherian (commutative) ring every irreducible ideal is primary. Can you give a counter example in a non noetherian ring? I've been lookin' ...
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1answer
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Prime ideals of Z

I must be going crazy... We know that for an integral domain, $R, a \in R$ is prime if and only if $(a)$ is a prime ideal. So taking $R$ to be the integers and $a=2$. Obviously 2 is prime and looking ...
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Find the number of prime ideals (CSIR 2014)

Let $p,q$ be distinct primes. Then (1) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct ideals. (2) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct prime ideals. (3) $\dfrac{\mathbb{Z}}{p^2q}$ ...
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1answer
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Ideals of a skew polynomial ring where no positive power of the automorphism is inner

The exercise I'm trying to answer is as follows: Let $R$ be a ring, and $\alpha : R \rightarrow R$ an automorphism of $R$. Suppose that $R$ is simple and that no positive power of $\alpha$ is ...
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Does $ax\in\mathfrak{m}I$ with $x\in I\setminus\mathfrak{m}I$ and $a \in R$ imply $a\in\mathfrak{m}$ for an invertible fractional $R$-ideal $I$?

Let $R$ be an integral domain, $\mathfrak{m}$ a maximal ideal of $R$, and $I$ an invertible fractional $R$-ideal. If $x \in I \setminus \mathfrak{m}I$ and $a \not\in \mathfrak{m}$, do we have $ax ...
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1answer
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Is the set of polynomials an $x^n+ a_{n-1}x^{n-1}+\ldots+a_1x +a_0$ such that $2^k+1$ divides $a_k$ an ideal in $\Bbb Z[x]$?

Is the set of polynomials $a_nx^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0$ such that $2^k+1$ divides $a_k$ an ideal in $\Bbb Z[x]$? I think it is true for $2^k+1$ and it will be true for all the divisors as ...
3
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1answer
51 views

Compute the transcendence degree (transcendence degree and tensor products)

$\DeclareMathOperator{\quot}{Quot}\DeclareMathOperator{\tr}{tr}$ Let $I_1$ and $I_2$ be nontrivial ideals in $\mathbb C[x_1,\ldots,x_k]$ and $\mathbb C[y_1,\ldots,y_m]$, respectively. Define $$ R_1 ...
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1answer
26 views

Example of ring with two maximal ideals such that the char of the quotients is $0$, respectively $p$.

I am looking either for an example of a commutative ring with identity and two maximal ideals, such that the characteristic of one of the quotient rings is finite and the other characteristic is zero, ...
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0answers
40 views

Meaning of $S^{-1}R$ notation

Here are objects defined in an exercise: Let $R$ be a commutative ring. Let $A$ be an ideal of $R$ and $S=\{1+a\mid a\in A\}$. The exercise then makes reference to the prime ideals of $S^{-1}R$. ...
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77 views

Prime ideals in $k[x,y]/(xy-1)$.

Let $k$ a field. Let $f$ be the ring injective homomorphism $$ f:k[x] \rightarrow k[x,y]/(xy-1)$$ obtained as the composition of the inclusion $k[x] \subset k[x,y]$ and the natural projection map $ ...
2
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1answer
40 views

Ideals, Dedekind domain and $\mathbb{Z}[\sqrt{-3}]$

I have the ideal $\mathfrak{a} = (2, 1 + \sqrt{-3})$ in $\mathbb{Z}[\sqrt{-3}]$. I have to show that $\mathfrak{a} \neq (2)$ but $\mathfrak{a}^{2} = (2)\mathfrak{a}$ and then conclude that ideals do ...
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1answer
89 views

In an extension of finitely generated $k$-algebras the contraction of a maximal ideal is also maximal

Let $k$ be a field and let $A \subset B$ be two finitely generated $k$-algebras. Prove that the contraction of any maximal ideal of $B$ is a maximal ideal of $A$. thank you very much again!