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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$K[x,y]$ (where $K$ is a field) have any bound for the number of generators of ideals?

We know that maximal ideals of $K[x_1,x_2,...,x_n]$ have $n$ generators. But is there any bound for the number of generators of arbitrary ideals? (For example in $K[x,y]$.)
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If $1\in M$, could $M$ be a maximal ideal of a commutative ring with identity $R$?

If $1\in M$, could $M$ be a maximal ideal of a commutative ring with identity $R$? I know that this is a very silly question. I think that the answer is that $M$ can't contain the identity for ...
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Conductor of a ring

An easy (possibly trivial) question from Neukirch's Algebraic Number Theory, p.47. Let $A$ be a Dedekind domain, $K$ its fraction field, $L$ a finite separable extension of $K$ and $B$ the integral ...
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27 views

properties of finitely generated torsion-modules and their submodules over a PID

Let $R$ be a principal ideal domain, $M$ a finitely generated $R$-torsion module, and $N \subseteq M$ a submodule. I want to show that there exist free R-modules $F, F', F''$ and module homomorphisms ...
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infinitely many ideals

does the ring $\Bbb Z_2[x]$ have infinitely many ideals like $\Bbb Z[x]$? How do you know if a ring has a finite number of ideal. particularly asking about seemingly large rings.
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Can we prove, without axiom of choice, that the set of all zero divisors (including $0$) of a commutative ring with unity contains a prime ideal?

Let $R$ be a commutative ring with unity , I know that assuming axiom of choice , if $A$ is the set of all zero divisors (including $0$ ) then it is a union of prime ideals so it contains a prime ...
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24 views

Ideal in ring product [closed]

Let $R$ and $S$ be two rings, then $K$ is an ideal in $R\times S$ if and only if there are $K_1$ and $K_2$ such that $K_1$ is an ideal of $R$ and $K_2$ is an ideal of $S$ such that $K = K_1 \times ...
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Maximal left ideals $\leftrightarrow$ simple left modules

Suppose $R$ is a ring with unity. This passage in Lang's Algebra discusses the correspondence $$\text{Maximal left ideals of $R$} \leftrightarrow \text{Simple left $R$ modules},$$ where I corresponds ...
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Question on intersection of ideals.

Consider the polynomial ring $k[x_0,x_1]$, and the two ideals $I=(x_0,x^2_0 x^2_1,x^3_1)$ and $J=(x^2_0,x^2_1)$. What is the intersection of these ideals? I found that $I \cap J = ...
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bijection between prime ideals of $R_p$ and prime ideals of $R$ contained in $P$

Given a ring $R$, I want to show that the localization of $R$ at the prime ideal $P$ of $R$(denoted as $R_P$) is isomorphic to the set of prime ideals of $R$ contained in $P$. That is: ...
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Finding primary decompositions of ideals

I have been given this example of the decomposition of an ideal into primary ideals $$ I =⟨x^2,xy,x^2z^2,yz^2⟩$$ Then the primary decomposition of this ideal is: $$⟨x^2,y⟩∩⟨x,z^2⟩⊆K[x,y,z]$$ This ...
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Irreducible decomposition of varieties vs primary decomposition of ideals

I'm new to working with varieties, and the statement mentioned below is left as an exercise, but I'm having some difficulty trying to prove it. Let $R=K[x_1,...,x_n]$. If $X=X_1\cup ... \cup X_n$, ...
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1answer
33 views

Show that the variety $V(I(X))=X$

In the ring $R=K[x_1,...,x_n]$, the variety of an ideal is defined as $V(I)=\{(a_1,...,a_n)\in K^n|f(a_1,...,a_n)=0, \space\forall f\in I\}$ The ideal of a variety is defined as $I(V)=\{f\in ...
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Questions about homomorphisms?

I'm running into algebra using homomorphisms for A LOT of things, but I don't think I have a full understanding of what they are. I have read on this site a good explanation that was really great, but ...
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41 views

Question about principle ideals and polynomials and quotient ring construction?

Say I have a ring of polynomials in $R[x]$. I wish to define the quotient group $R[x]/<x^2+1>$. My question lies in the ideal generated by $<x^2 + 1>$. This is the set of all numbers such ...
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Question about maximal ideals in a Polynomial ring

I'm reading Freiligh and he has an example in a book, here it is: Let $F = R$ and let $f(x) = x^2 + 1$. Which is well known to have no zeros in $R$ and thus is irreducible over $R$ by a theorem ...
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Solving Mordell Equations

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
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For a group ring, finding if a subset is an ideal. [closed]

For the ring $R=SG$, the group ring of a finite group G over an integral domain S, and a subset $I=(g-1|g \in G)$, is this subset an ideal? Is it prime? How about maximal?
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Showing a Ring and an Ideal are equal

Q: Let R be a commutative ring with unity. Prove that if A is an ideal of R and A contains a unit, then A=R. This is my attempt at an answer: It suffices to show that all the elements in R are in A. ...
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Proving that the three statements on units are equivalent

Let $R$ be a ring and $x\in R$ i) $x$ is a unit ii) $\bar{x}$ is a unit in $R/P$, where $P$ is a prime ideal iii) $\frac{x}{1}$ is a unit in $R_P$, where $P$ is a prime ideal i)$\implies$ ii) if $x$ ...
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1answer
38 views

how to tell if a ring is noetherian

In general how would i tell if the rings $\mathbb{Z}[\sqrt d]$ and $\mathbb{Z}[\frac{x}{y}]$ are noetherian? I know that the ring $\mathbb{Z}$ is noetherian as all ideals are contained in a finite ...
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Ideals in $\mathbb{Z}[X]$ with three generators (and not with two) [duplicate]

It is well-known that in $\mathbb{Z}[X]$ we do have non-principal ideals, for example $(2,x)$. This is an ideal with two generators. Now I was wondering if there exists an ideal with three generators, ...
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Ideals in $\mathbb{C}[x,y]/I$ where $V(I)$ is finite set.

$R = \mathbb{C}[x,y]$, $I \subset R$ ideal and $V(I)$ - is finite set. I want to prove that all prime ideals in $R/I$ are maximal. I know that it's true for $R/radI$. Let $\pi$ be natural map $R/I ...
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1answer
49 views

Showing an ideal is irreducible

I am currently trying to show that the ideal $\langle x^3, y^5, z^2 \rangle \subset \mathbb{C}[x,y,z]$ is irreducible (i.e.: it cannot be written as the intersection of two larger ideals $J$ and ...
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1answer
40 views

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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1answer
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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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34 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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4answers
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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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1answer
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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1answer
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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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1answer
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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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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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1answer
48 views

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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1answer
18 views

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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131 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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votes
2answers
49 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
40 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 ...