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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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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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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$\overline{\mathbb{Z}}$ 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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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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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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63 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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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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20 views

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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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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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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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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65 views

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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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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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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25 views

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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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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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 ...
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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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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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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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29 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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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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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 $ ...
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A monomorphism from a ring to a direct sum

Let $R$ be a ring with a family of ideals $A_i$'s ($i\in I)$. We could consider a well-defined $R$-monomorphism from $R/∩A_i$ to the direct product of $R/A_i$'s sending $r+∩A_i$ to the tuple ...
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Finitely generated prime ideal and annihilator

Suppose $R$ is a commutative ring, $P$ is a prime ideal of $R$, $P$ is finitely generated, and $\operatorname{Ann}(P)=0$. Show that $$\operatorname{Ann}(P/P^2)=P.$$ These are my efforts: ...
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A relation in a finitely generated module

Suppose $R$ is a commutative ring, $I$ is an ideal of $R$, and $M$ a finitely generated $R$-module s.t. $M=IM$. How to prove: $$\exists a \in I \text{ such that } (1-a)M=0. $$ I tried to solve: ...
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Rings and modules

Let $R$ be a ring in which every maximal ideal is a direct sum of cyclic $R$-modules. Now let $I$ be a proper ideal of $R$. What is the structure of $I$. Is it true that $I$ is a direct sum of cyclic ...
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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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Which of the following is also an ideal?

If $U,V$ are ideals of a ring $R$, then which of the following is also an ideal of $R$? $U+V=\{u+v\mid u\in U,v\in V\}$ $U\cdot V=\{u\cdot v\mid u\in U,v\in V\}$ $U\cap V$ My attempt: I have ...
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A question from the Herstein's “noncommutative ring”, page 15, theorem 1.2.5

Let $R$ be a ring. $\rho$ is a maximal regular right ideal of $R$. $(R/\rho,+)$ is an irreducible $R$ module. $(r+\rho)\cdot r_1$ is defined by $rr_1+\rho$. $A$ is a two-sided ideal of $R$. ...
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If $P \in \operatorname{Ass}M$, then $R/P \subset M$.

Let $R$ be a commutative ring with unity. $M$ an $R$-module. Then $P \in \operatorname{Ass}M$ if and only if there is a submodule $N\subset M$ such that $R/P \cong N$. ...
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what does 2 ideals are equal mean?

I'm revisiting the proof of 1-1 correspondence theorem and while proving $f$ is one-one I don't know how to write mathematically what we mean by 2 ideals are equal? (Here $f$ is a map from set of ...
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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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42 views

1-1 correspondence theorem

Here is the correspondence theorem stated as follows: Let $A$ be an Ideal of ring $R$.There is 1-1 correspondence between Ideals of $B$ containing $A$ and ideals of $R/A$. I have read the proof but ...
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Generators of an ideal in rings of power series

Please help me for solving a homework. Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that ...
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How to find the number of maximal ideals? [closed]

Let $n \geq 2$ and $n={p_1}^{e_1}{p_2}^{e_2}\cdots {p_r}^{e_r}$. Then the number of maximal ideal of $Z/nZ=Z_n$ is r n $e_1$+$e_2$+....+$e_r$ $p_1$$p_2$....$p_r$
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Relation between ideals in Noetherian domains.

Suppose that we have a Noetherian domain $R$ and two ideals $I$ and $J$ of $R.$ Now consider the minimal (or irredundant) primary decompositions $I=\bigcap\limits_{i=1}^r Q_i$ and ...
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Determining consistency of a general overdetermined linear system

For $m > 2$, consider the $m \times 2$ (overdetermined) linear system $$A \mathbf{x} = \mathbf{b}$$ with (general) coefficients in a field $\mathbb{F}$; in components we write the system as ...
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Showing an ideal with maximality condition is prime.

Let $R$ be a commutative domain and suppose that $I \subseteq R$ is an ideal of $R$ maximal with respect to the property that $I^{-1} \not\subseteq R$. Show that $I$ is a prime ideal. This is ...
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Finding the element of the quotient ring $Z[i]/\langle 2+2i\rangle$

First, I'm writing an element to confirm whether I understood this quotient ring correctly. $$(5 + 7i) + \langle 2+2i \rangle = 2(2+2i) + (1+3i) + \langle 2+2i \rangle = ...
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For any two Ideals $A$ and $B$,$A+B=\langle A \cup B \rangle$

Below is the proof of : Prove that for any two ideals $A$ and $B$ of ring $R$,$A+B=\langle A \cup B~\rangle$ . Proof: By theorem (for any two ideals of a ring $R$ ,then the set $A+B$ is an ...
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Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
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Sets of prime ideal contain a minimal element

I want to prove that every nonempty set of prime ideal contain a minimal element, my attempt is to prove it by using zorns lemma and i would like to know if my proof is valid. Let $\Sigma$ be a ...
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Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
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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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Prove that, if $(a)=(a')$, then $a'=ua$

Let $R$ be integral domain. Show that if $2$ principal ideals $(a)$ and $(a')$ are equal (where $a,a'\in R$) then there exists $u\in R^{\times}$ such that, $a'=ua$ Now if $(a)=(a')$ then ...
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Radical ideals of $\mathbb{Z}$?

I am having trouble with classification of the radical ideals of $\mathbb{Z}$. We know that for a commutative ring $R$ with an ideal $I$, the radical of $I$ is defined (and denoted as $\sqrt{I}$) as ...
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Non Maximal Prime ideal! [duplicate]

Assume $S$ to be all continuous functions from $[0,1]$ to $\mathbb R$. I know by compactness of $[0,1]$ it follows that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$.Does ...