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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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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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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ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
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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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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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Isomorphism and Quotient Ring [on hold]

Let $R$ be a ring. If for every proper ideal $I$ of $R$ we have $R/I\cong R$, then show that for every two proper ideals $I$ and $J$ of $R$ either $I\subseteq J$ or $J\subseteq I$.
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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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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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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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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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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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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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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 ...
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if the sum of two units is a unit, then there is an unique maximal ideal

Let $R$ be a ring with identity element. I have to proof that if the sum of two units of $R$ is a unit, then $R$ has an unique maximal ideal. But i don't see a connection. If someone could give me a ...
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Quotient of the ring of integers of a quadratic field by the ideal generated by a split integer prime.

I am wondering about primes $p$ in $\mathbb Z$ that are split in $\mathcal O_{K}$, $K=\mathbb Q(\sqrt d)$. Let $\omega=\sqrt d$ if $d \equiv 2,3 \mod 4$ and $\omega=\frac{1+\sqrt d}{2}$ if $d \equiv 1 ...
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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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Factoring the Ring of Integers into Ideals

Let $K$ be a number field. Let $\frak p$ be a prime ideal in $\mathcal O_K$. Let $u\in \mathcal O_K$ and $m\in \mathbb N$. I've been told that $|u|_{\frak p} = |m|_{\frak p} = 1$ where $|\cdot|_{\frak ...
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How to prove that an ideal can not be generated by 2 elements

In Kunz's "Introduction to commutative algebra and algebraic geometry", page 137-139, particular monomial affine curves are described. Here is the link. In case the curve is not an ideal ...
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Factorising ideals in the ring of integers of a quadratic field

In an undergraduate algebraic number theory course, I was given the question "If $K = \mathbb Q(\sqrt{-33})$ Factorise the ideal $(1+\sqrt{-33})\subset \mathcal O_K$ into a product of prime ideals." I ...
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how to show that an ideal is convex [closed]

I need to show that the ideal $J=(i)$ in $C(\mathbb R)$ where $i$ is the identity function, $C(\mathbb R)$ is the ring of all continuous functions on the real numbers, is a convex ideal.
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What are some examples of principal, proper ideals that have height at least $2$?

Krull's principal ideal theorem states that in a Noetherian ring $R$, any principal proper ideal $I$ has height at most $1$. Presumably the Noetherian hypothesis is required, so what are some ...
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The difference between the ring version and module version of Chinese Remainder Thereom.

Chinese Remainder Theorem for Commutative Rings If $R$ is a commutative ring with $1$ and $I, J$ are ideals of $R$ that are pairwise coprime or comaximal (meaning $I + J = R$), then $IJ = I \cap J$, ...
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Why $R/AB$ is cyclic?

Why $R/AB$ is cyclic when $A,B$ are ideals of $R$? I know a cyclic module is a module that can be written as $Rm$ and $Rm$ is isomorphic to $R/Ann(m)$. However, I can't see why the module is ...
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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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Are ideals also rings?

I am learning about rings and ideals. But I am confused about something. My book (Gallian) says that an ideal of a ring by definition is a subring. But I have talked to other people who insist that an ...
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Quotient of Ideals in matrix rings

I'd like to know where could I find some info about the quotient $I:J=\{a\in R\mid aJ\subseteq I\}$ ($R$ a ring) in matrix rings? Or for example, in a matrix ring over $\mathbb{Z}$. I would like to ...
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Let $R$ be a PID and $I$ a prime ideal of $R$ s.t. $0 \subset I \subset 1_R$

Let $R$ be a PID and $I$ a prime ideal of $R$ s.t. $0 \subset I \subset 1_R$ and let $I = \langle a \rangle$, where $a$ is a prime element of $R$. My question is: is there any other prime ideal $J$ ...
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Prove that $\Bbb{Z}[i]/I$ is finite where I is an ideal of $\Bbb{Z}[i]$

Show that for any nontrivial ideal $I$ of $\Bbb{Z}[i]$, $\Bbb{Z}[i]/I$ is finite. $\Bbb{Z}[i]$ is a PID, so $I=\langle{a+ib\rangle}$. Now $\Bbb{Z}[i]/I$ has elements of the form ...
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$A$ prime in $S$ implies that $\phi^{-1}(A)$ prime in $R$ ; $A$ maximal in $S$ implies that $\phi^{-1}(A)$ maximal in $R$

Suppose $R,S$ are commutative rings with unities. Let $\phi$ be a ring homomorphism mapping $R\to S$ and let $A\subset S$ be an ideal. How can I start the proofs for: Showing that $A$ prime in ...
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Maximal ideals of the ring of matrices

Let K be a field. We consider $K^n$ as a left module of $M_{n, n}(K)$, the ring of matrices of size $n$ over $K$. 1) For any $M_{n, n}(K)$ module homomorphism $ 0 ≠ \phi: M_{n, n}(K) \to K^n$, show ...
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Finitely generated modules and submodules

Let $R$ be a ring, $M$ an $R$-module and $U$ a submodule of $M$. Show that, if $U$ and $M/U$ are finitely generated, then $M$ is finitely generated aswell. I thought to maybe show this by taking ...
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Is $ \langle x,5 \rangle $ a maximal ideal of $ \mathbb{Z}[x] $?

Here, $ \langle x,5 \rangle $ is the ideal generated by $ x $ and $ 5 $ in $ \mathbb{Z}[x] $, which is the polynomial ring over $ \mathbb{Z} $. How should I approach this question?
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Proving that a Left semisimple ring $R$ is both left noetherian and left artinian

Prove that a left semi-simple ring $R$ is both left noetherian and left-artinian. I am following the proof given in pg 27,A first course in non-commutative rings (T.Y.Lam). Its strategy is to show ...
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If C is a chain of non-principal ideals and the union of the ideals of the chain contains a generator, why is the union then principal

If C is a chain of non-principal ideals and the union of the ideals of the chain contains a generator why is the union then principal. I understand this is a contradiction already and seems painfully ...
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Relatively prime ideals in Dedekind Domains

I am currently working through Lang's Algebra and have come across an exercise I can not solve (Chapter II, Exercise $19$). Any help would be greatly appreciated. Let $R$ be a Dedekind domain. ...
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Cardinomials: Like cardinalities, but polynomial valued

I want to see if this notion is known (or if it makes sense). Let $F$ be a field. Let $A$ be a finite dimensional commutative unital algebra over $F$. Let $X_1$, $X_2 \in A$ etc. be such that their ...
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Is it true that the order of any quotient ring $\mathbb Z[i]/\langle a+ib \rangle $ is $a^2+b^2$ ? (where not both $a,b$ are zero)

Is it true that the order of any quotient ring $\mathbb Z[i]/\langle a+ib \rangle $ is $a^2+b^2$ ? ( I know it is atmost finite ) Please help . Thanks in advance .
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Let $I = \{a+ ib \in \Bbb Z[i] : 2 \mid a-b\}$ then $I$ is a maximal ideal of $ \Bbb Z[i]$.

Let $I = \{a+ ib \in \Bbb Z[i] : 2 \mid a-b\}$ then $I$ is a maximal ideal of $ \Bbb Z[i]$. We consider an ideal $J$ such that $I \subset J\subset\Bbb Z[i] $. So there exists an element $p \in J$ but ...
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I've proved everything about the ideal correspondence easily except $\pi ^{-1} \pi (\frak{a}) = \frak{a}$

The correspondence theorem to which I refer is the bijection between ideals of a commutative ring with $1$, $A$, and ideals of $A/\frak{b}$. I can prove easily most parts that imply the bijection ...
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Let I be an unmixed radical ideal of R. then (I:x) is unmixed

Let $R$ be commutative ring with $1$. One says that an ideal $I$ is unmixed if $I$ has no embedded prime divisors (in other words, if the associated prime ideals of $R/I$ are the minimal prime ideals ...
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How can I verify that the ideal $(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$ in $\mathbb Q[x,y,z,w]$

I want to show that the ideal $$(x^2-zw, z^2-yw, y^3-xw, w^3-xy^2z)$$ in the ring $\mathbb{Q}[x,y,z,w]$ is prime, how can I?
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Is it true that an integral domain $R$ is a UFD if and only if intersection of any two principal ideals of $R$ is principal ?

Is it true that an integral domain $R$ is a UFD if and only if intersection of any two principal ideals of $R$ is principal ?
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If every maximal ideal is finitely generated is the ring Noetherian? [duplicate]

$R$ is a commutative ring with $1$. Suppose every maximal ideal is finitely generated. Is this ring Noetherian? Equivalently, is every prime ideal finitely generated?
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Are all prime ideals in $\mathbb Z[\sqrt{-5}]$ of the form $\langle c, a + b \sqrt{-5} \rangle$?

Where $a, b, c \in \mathbb Z$? I know that if in an UFD, $\langle c, a + b \sqrt{d} \rangle$ would boil down to a principal ideal. But it seems to me that in $\mathbb Z[\sqrt{-5}]$, for any purely ...
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Let $R$ be a PID. Prove that $\exists c \in R$ such that $c\mid a, c\mid b$ and $c = ax + by$.

Let $R$ be a PID and $a,b \in R$. Prove that $\exists c \in R$ such that $c\mid a, c\mid b$ and $c = ax + by$ for some $x,y \in R$.
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True or false question about polynomial ring

Let, $\mathbb{R}[x]$ be a polynomial ring and let $J = (x)$. True/false: $J$ consists of all the polynomials of $\mathbb{R}[x]$ whose constant terms are $0$. I know $J=(x)$ is a maximal ideal of ...
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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 ...