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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Rings, ideals and quotient rings

Suppose $I$ is an ideal of ring $R$, and $J'$ is an ideal of $R/I$. Show there is an ideal $J$ in $R$ so that $J/I=J'$. How do I answer this? What am I required to prove?
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Trouble finding the norm of the two following ideals

Given that $\alpha$ is the root of the polynomial, $x^3 - x - 1$ is $\alpha$ and $K=\mathbb{Q}(\alpha)$, show that the norm of the ideal $\langle 5, \alpha-2\rangle$ is $5$ and the norm of the ideal ...
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Varieties and ideals

I'm doing the exercises from Fulton of Algebraic Geometry and I'm stuck in the problem 2.44 Let $V$ be a variety in $\mathbb{A}^{n}$, $I=I(V)\subset k[x_{1},\ldots,x_{n}]$, $P\in V$ and let $J$ be ...
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35 views

Intersection and isomorphism of two relatively prime ideals

Let $R$ be a commutative ring. Using the definition that two ideals $I, J \subseteq R$ are relatively prime if $I + J = R$. I want to show that for two relatively prime ideals $I, J \subseteq R$, it ...
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How to show an ideal is principal

Is there a general procedure to check whether or not a prime ideal of the ring of integers $O_K$ is principal. In my case $K$ is a quadratic field, i.e $\mathbb{Q}(\sqrt {d})$, with $d$ square-free.
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For a maximal left ideal $M$ of $S$, is $f^{-1}(M)$ a maximal left ideal of $R$ when $f$ is surjective?

Let $f:R \longrightarrow S$ a surjective ring homomorphism. Is the inverse image $f^{-1}(M)$ a maximal left ideal of $R$ for any maximal left ideal $M$ of $S$? Comments: I tied something ...
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Why does $N(\mathfrak p)$ belong to $\mathfrak p$?

Why does $N(\mathfrak p)$ belong to $\mathfrak p$ ? $N(\mathfrak p)$ is the norm of the prime ideal $\mathfrak p\in\mathcal O$ defined as $N(\mathfrak p)=|\mathcal O/\mathfrak p|=$(say $p$) Now ...
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Question about maximal ideals?

I'm reading Freligh and introduction to Abstract algebra and I'm getting confused. The set generated by $\langle x^2 + 1\rangle$ is a maximal ideal in $R[x]$. First, I don't understand it. $\langle ...
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35 views

subring of a quotient field

Let $R$ be a principal ideal domain, and $S \subseteq Q(R)$ a subring of the quotient field of $R$, so that $R \subseteq S$. I want to show that, for any $x, y \in R$: $$\frac{x}{y} \in S \implies ...
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25 views

Why $\mathfrak p_2\cdots\mathfrak p_r\not\subset (a)\mathcal O$

If $(a)\mathcal O\subset\mathfrak p_1$ and $r$ is the minimal number such that $\mathfrak p_1\cdot\mathfrak p_2\cdots\mathfrak p_r\subset (a)\mathcal O$ then $\mathfrak p_2\cdots\mathfrak ...
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Proof that every Principal ideal domain is Noetherian [duplicate]

I would like to know if my logic is sound. We know that in every principal ideal domain, every ideal is multiplicatively generated. Thus, for $a \in R$ we have: $aR = ${$ra: r \in R$} Thus every ...
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25 views

For fractional ideal $I$ why is $I\cap R \supsetneq \{0\}$?

In the proof in my textbook that a fractional ideal $I$ in a quotient field $K$ of an integral domain $R$ has an inverse $$I^{-1} = \{ x\in K : x I \subseteq R\}\,,$$ it is used that there exists an ...
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Simple examples of fractional ideals

Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ not $\{0\}$, for which a $0 \neq r \in R$ exists so that $r I \subseteq R$ is an ideal in $R$. ...
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Hints on how to approach a problem concerning rings/field in Abstract Algebra

I am a student, prepping for a final exam in graduate Abstract Algebra. My professor has told me that he will be giving us the following two problems in class to turn in: (1) Given that R is an ...
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An example of a ring R and a subring R' with R' not an ideal of R

And also, another thing, I'm curious about. They say that an Ideal is the analogue of the Normal subgroup in group theory, but that confuses me. Let a Group be G. Let a subgroup be H. H is normal in ...
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Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal.

Show that the ideal generated by $4$ in $\mathbb Z_{12}$ is not a prime ideal. Hint: Give a counter-example This is my rough proof to this question. I was wondering if anybody can look over it ...
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On general topological spaces and $C(X, \mathbb R)$ , where for closed sets $A,B$ in $X$ , $I_A=I_B \implies A=B$

Let $X$ be a metric space and $C(X, \mathbb R)$ be the ring of all real valued continuous functions from $X$ . For $A \subseteq X$ , let us define $I_A :=\{f \in C(X, \mathbb R) : f(x)=0 , \forall x ...
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factor out of an expression, a couple principal ideals, software?

I have an expression, $f$, consisting of a few rational fractions of large multivariate numerators, $n1,\,n2,\ldots \in \mathbb{Q}[a1,a2,b1,b2;Q]$ and large multivariate denominators, $d1,\,d2,\ldots ...
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Let $I = (x^2, y)$ be an ideal of $\mathbb{Q}[x,y]$ show that Rad$(I)$ = $(x,y)$ and $I$ is a primary ideal that is not a power of a prime ideal

Let $I = (x^2, y)$ be an ideal of $\mathbb{Q}[x,y]$ show that Rad$(I)$ = $(x,y)$ and $I$ is a primary ideal that is not a power of a prime ideal. I can see that $(x,y) \subset Rad(I)$ b/c $x^1, y^2 ...
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21 views

Prove $R$ conatins an ideal that is not finitely generated. $R = F[x,x^2 y,\ldots,x^n y^{n-1},\ldots]$

Prove R conatins an ideal that is not finitely generated. $R = F[x,x^{2}y,\ldots,x^n y^{n-1},\ldots]$ and is a subring of $F[x,y]$ where $F$ is a field. Seems like $R$ itself is not finitely ...
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A radical ideal in a commutative ring is prime if and only if it is not an intersection of two radical ideals properly containing it?

Let $I$ be a radical ideal (i.e. $\sqrt I=I$) in a commutative ring with unity. Then is it true that $I$ is a prime ideal if and only if it is not an intersection of two radical ideals properly ...
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Is the image of a ring homomorphism an ideal?

Let $\phi : R \rightarrow S$ be a ring homomorphism, then is $im(\phi)$ an ideal in $S$? I ask this because I am studying about modules and in that we say that for a given $R$-module homomorphism the ...
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132 views

Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Prove the following:

Let $R$ be a commutative ring with $1 \ne 0$, and let $0 \ne e \in R$ be an idempotent element. Note that $eR=\{er|r \in R\}$ is also a commutative ring with identity element $e$. (1) If I is an ...
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54 views

Form of maximal ideals in an algebraicaly closed polynomial ring

I have been trying to prove the following bijection which is a consequence of the nullstellansatz $$\{\text{maximal ideals of }\mathbb{C}[x_1,\dots,x_n] \} \leftrightarrow \{\text{points in ...
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2answers
44 views

An example of an ideal of order $12$

Provide an example of an ideal in $R=\mathbb{Z}_6\times\mathbb{Z}_{10}$ that has order $12$, and indicate whether the ideal is a principal ideal (if it is, then identify the generator for the ...
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1answer
36 views

Find a counterexample to the following lemma if we change the statement slightly.

let K be an algebraic number field and let $O_K$ be its ring of integers. Lemma; Let $a,b$ be fractional ideals of $O_K$. If $b \subseteq a$ then there is an ideal $c$ such that $b=ac$. I need to ...
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What is the number of elements of $\mathbb Z[i] /I $, where $I:=\{a+ib \in \mathbb Z[i] : 2 \mid a-b\}$?

I know that $I:=\{a+ib \in \mathbb Z[i] : 2\mid a-b\}$ is a maximal ideal of $\mathbb Z[i]$. My question is: what is the number of elements of $\mathbb Z[i] /I $? I am totally stuck. Please help ...
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49 views

Power series with coefficients in primary ideals

Let $P$ be a prime ideal in a commutative ring $R$ with unity such that an ideal $Q$ is $P$-primary and some power of $P$ is a subset of $Q$. I want to show that $\sqrt {Q[[x]]}=P[[x]]$. If a ...
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2answers
37 views

Pre-image of a ring homomorphism

Let $\phi:R \rightarrow S$ be a ring homomorphism and let $J$ be an ideal in $S$, then it is quite easy to prove that $\phi^{-1}(J)$ is an ideal in $R$. But can someone help me with proving the ...
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Can a local ring have more than one prime ideal?

A local ring is defined as a ring which has a unique maximal ideal. This unique maximal ideal consists of only non-units and contains all the non-units of the ring $R$. So examples of local rings ...
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Prime Ideals in a Field

This is probably a really simple question: Am I right in thinking that a field has only one prime and one maximal ideal, namely the $(0)$ ideal? I think this is so because both prime/maximal ideals ...
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Ideal generated by two polynomials

Let a sequence of polynomials $\{f_n\}_{n=0}^\infty$ in $\mathbb{Q}[x,y]$ be given in the following way: $$f_0=1,$$ $$f_1=-x,$$ $$f_2=x^2-y,$$ $$f_{n+2}=-xf_{n+1}-yf_n.$$ For each $n\geq 0$, find ...
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1answer
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Prime Ideals of a ring

The definition of a prime ideal is: An ideal $I$ in a ring $R$ is said to be a prime ideal if $I \neq R$ and $f.g \in I \implies f \in I$ or $g \in I$ My understanding of this is that $f.g \in I$ ...
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Finding the left (or right) ideals of the ring of $n\times n$ matrices

Just give me a hint, since this is assessment! DO NOT TELL ME THE IDEAL I want to find the left (or right) ideals of the ring of $n\times n$ complex valued matrices. Now the definition is (for ...
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27 views

Question on the product of ideals

Let $R$ be a commutative ring. Let $P,Q$ be two ideals of $R$. Then I know that $PQ$, the product of ideals, is the ideal generated by $pq$ for each $p\in P$ and $q\in Q$. In other words, ...
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37 views

$\varphi : R → S$ is a epimorphism from $R$ to ring $S$, let $I$ be an ideal of $R$. Prove $\varphi (I) = S$ if and only if $R = I +Ker(\varphi)$

Let $\varphi : R → S$ be an epimorphism from ring $R$ to ring $S$, and let $I$ be an ideal of $R$. Prove that $\varphi (I) = S$ if and only if $R = I +Ker(\varphi)$ I am quite confused on what ...
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1answer
59 views

Computing class group of $\mathbb Q(\sqrt{6})$

I am calculating the class group of $\mathbb Q(\sqrt 6)$. My working is as follows: The Minkowski bound is $\lambda(6)=\sqrt 6<3$ so we only need to look at prime ideals of norm $2$. $2$ divides ...
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1answer
52 views

Least common multiple for integer matrices

Given two full-rank $3\times3$ integer matrices $M_1$ and $M_2$, I am trying to find integer matrices $N_1$ and $N_2$ such that $M_1N_1$=$M_2N_2$, such that $\left|\det(M_1N_1)\right|$ is minimal. ...
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Quotient of graded rings by graded ideals are again graded rings

I want to prove the following statement. Let $A = \bigoplus_{n=0}^{\infty}A_n$ to be a graded $R$-algebra and $I$ a graded ideal of $A$. Let $(A/I)_n = (A_n + I)/I$ be the image of $A_n$ in $A/I$. ...
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Two questions concerning ideal factorization and norm

$\bullet$ In $\mathbb Z[\sqrt{-5}]$ why is $(2)=(2,1+\sqrt{-5})(2,1-\sqrt{-5})$ Actually both ideals on the RHS contain $(2)$, but also their product ? Can we just multiply RHS in the normal sense; ...
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27 views

Ideals and Null set

I was wondering if an Ideal in a ring can ever be the null set. The definition of an Ideal $I$ is that it is a subset of the ring $R$ such that: 1)It is an abelian group under "addition" (I put it in ...
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1answer
32 views

What is the difference between these two conditions $J = \{az \mid a \in R\}$ and $ I = \{a \in R \mid az \in J\}$

Please consider these two questions: Let $R$ be a ring and $z \in R$, which is fixed. Let, $J = \{az \mid a \in R\}$. Prove that $J$ is a left ideal of $R$. Skipping the subtraction part, this is ...
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Show if $2$ and $1+\sqrt{-5}$ belong to the same principal ideal $I$ of $\mathbb{Z}[\sqrt{-5}]$ then $I=\mathbb{Z}[\sqrt{-5}]$.

Show if $2$ and $1+\sqrt{-5}$ belong to the same principal ideal $I$ of $\mathbb{Z}[\sqrt{-5}]$ then $I=\mathbb{Z}[\sqrt{-5}]$. I have proved so far that 2 and $1+\sqrt{-5}$ is irreducible and ...
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2answers
103 views

On the minimal set of generators of monomial ideals in $\mathbb{C}[x,y]$.

I am trying to do exercise 2.6 of Hassett's "Introduction to algebraic geometry": i) Give an example of a monomial ideal $I\subseteq\mathbb{C}[x,y]$ with a minimal set of generators consisting of ...
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1answer
24 views

The annihilator induces a module [duplicate]

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
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Question about the deduction of the quotient ring $R/I$

Yesterday we deduced on class how quotient groups were deduced and well defined. Let $R$ be a ring and $I$ an ideal of $R$. My professor proved us that the multiplication operation $$R/I \times R/I ...
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63 views

Example of ideal generated by two elements

I have an easy example on my notes that I don't understand. My teacher said that in $\mathbb{Z}$, $(2,3)=2\mathbb{Z}+3\mathbb{Z}$ is a principal ideal, because $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$. ...
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1answer
26 views

Simple modules over R isomorphic to R/I

Let $R$ be a ring, and let $M$ be a simple $R$-Module, meaning that it only has the trivial submodules {0} and $M$. Show that there's a maximal ideal $I \subset R$ so that $M \cong R/I$. Thanks in ...
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1answer
32 views

Maximal ideals of finite algebra over a local ring

Let $R$ be a local ring with residue field $k$. Let $A$ be an $R$-algebra which is finitely generated as $R$-module. I want to show that the maximal ideals of $A$ are in one-to-one correspondence ...
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1answer
48 views

Closed points are dense in $\operatorname{Spec} A$

From 3.6.J in Vakil: Let $k$ be a field, and let $A$ be a finitely generated $k$-algebra. We want to show the closed points are dense in $\operatorname{Spec} A$. This is the set of prime ideals of ...