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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Maximal ideal space of $C^*$-algebra of Riemann integrable functions

Let $R([0,1])$ be the unital commutative $C^*$-algebra of complex valued Riemann integrable functions on $[0,1]$ with pointwise operations and the supremum norm. In the 1980 paper The Gelfand space ...
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Two ideals both alike in dignity, in fair Paris where we lay our scene.

Let $A$ be an integral domain. I have to show that two ideals $\mathfrak a$ and $\mathfrak b$ are isomorphic as $A$-modules if and only if there exist $a$ and $b$ such that $a\mathfrak b=b\mathfrak ...
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202 views

Irreducible homogeneous ideals

I have the following question: Let $I$ be a homogeneous ideal. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideals? So, is it ...
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Counterexamples to the avoidance lemma for arbitrary ideals

Let $A$ be a commutative ring with $1$. Let $I$ and $J_k$, $k=1,\dots,n$ be ideals of $A$ with $I\subseteq \cup _{k=1}^n J_k$. Then I have obtained the following: (1) If $J_k$, $k=1,\dots,n$, are ...
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195 views

Non-cyclic unit groups

Is there any way to motivate why certain factor rings of $\mathbb{Z}, \mathbb{Z}[i]$, etc., to a prime power have non-cyclic unit groups? For example, the only such non-cyclic unit groups of factor ...
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Where in the proof did Herstein use the fact that $A$ is a two-sided ideal of $R$?

I'm reading Noncommutative Rings by I. N. Herstein. The theorem I'm having trouble with is 1.2.5, on page 16 of the book. Some definition 1. Regular ideal An ideal $\rho \subset R$ is called a ...
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149 views

Primary decomposition of ideals

How to find a primary decomposition of the ideal $I = (X^2, XY, XZ, YZ)$ in the ring $k[X,Y,Z]$? Is there a general rule for finding primary decompositions? Also how to show that $(X,Y)^{308}$ is a ...
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105 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
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Need help with finding generator

$I=\{a+bi \in R\mid a \equiv b\pmod 2\}$ is ideal of $R=Z[i]=\{a+bi\mid a,b \in Z\}$. Can somebody help me to find the generator of $I$?
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On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
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When are all (prime) ideals of an $R$-algebra, extensions of (prime) ideals of $R$?

Let $f:R\rightarrow R'$ be a homomorphism of commutative noetherian rings. When are all (prime) ideals of $R'$ extensions of (prime) ideals of $R$? Is it true for the case $R'$ is $R$-flat?
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Isomorphism of K-algebras

Let $k[x]$ be a polynomial ring and $I$ an ideal (resp. graded). If $k\subset K$ is a field extension, then prove that there is a natural (resp. graded) isomorphism of $K$-algebras: $$K[x]/IK[x]\cong ...
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57 views

About quotient ring

I want to find the value $|\mathbb{Z}({\sqrt{2})/(3+\sqrt{2})}|,|\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}|$ and also the number of ideals of $\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}$. But still not ...
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174 views

Ideals in Gaussian integers

Let $R:=\mathbb{Z}[i]$. Prove that every nonzero prime ideal $\mathfrak{P}$ of $R$ belongs to one of the following families: 1) $\mathfrak{P}=(1+i)R$ 2) $\mathfrak{P}=(a+bi)R$ where ...
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Proof about affine varieties

Ok so I have that $k$ is algebraically closed and $F$ is an element of $k^n$, and it is a reduced polynomial. We have that $V = V(F)$. In the book it says prove that $F$ generates $I(V)$ but in my ...
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164 views

In a PID without unit an ideal is maximal iff it is prime.

As the titles says, I need to show that in a PID $R$ an ideal is maximal iff it is prime. This is easy to do if $R$ has a multiplicative identity. I can not do it if $R$ does not have an identity. It ...
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390 views

Generators for the intersection of two ideals

Let $I=\langle a_1,\dots, a_s\rangle, J=\langle b_1,\dots, b_t\rangle$ be ideals of arbitrary commutative ring. Then we know that $I+J=\langle a_1,\dots, a_s, b_1,\dots, b_t\rangle, ...
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Finding ideal representatives in the class group of $\mathbb{Q}(\zeta_{23})$

I know that $\mathbb{Q}(\zeta_{23})$ has class number 3, and I am wondering how I can find ideal representatives of the two nonprincipal classes in the class group. I have tried looking at examples ...
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111 views

Reduced Gröbner basis and extension of scalars

Consider a field extension $L\subseteq K$, and let $\mathfrak a\neq 0$ be an ideal of the polynomial ring $L[T_1,\ldots,T_n]$. Suppose that a monomial order is fixed, so there exists a unique reduced ...
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Dedekind's criterion clarification

Dedekind's criterion gives a way of factoring $p\mathcal{O}_K$ into prime ideals. (See http://math.stanford.edu/~conrad/154Page/handouts/dedekindcrit.pdf) Is it true that the prime ideals ...
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Lattice with $3$ operations.

If $R$ is a commutative ring and $\mathcal I(R)$ denotes its set of ideals then I know that $\mathcal I(R)$ can be looked at as a complete lattice with intersection $I\cap J$ and addition $I+J$ as ...
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65 views

Problem about Gröbner basis.

I'd really appreciate if someone could help me. The problem is the following: If $\psi_1,...,\psi_m \in k[x_1,\dots,x_n]$ and consider the $k$-algebra homomorphism: ...
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Ideal of a Vanishing set $I(V(F[X,Y]))$ and how to repeat the computation.

The video I am getting this from is found here: https://www.youtube.com/watch?v=spHxUPvrkXw, it is around 5 minutes in. The first part of the question is: for $F[X,Y] = Y^2 - X^3 = 0$ find ...
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58 views

Example of irreducible ideal which is not strongly irreducible

I have read a paper with title Ideal Theory in Commutative Semirings by Reza Ebrahimi Atani and Shahabaddin Ebrahimi Atani. In this paper we have the following definitions: An ideal I is irreducible ...
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52 views

Normal ring and unmixed ideals

Let $R$ be a commutative Gorenstein local ring , $I$ an ideal of $R$ . If $R/I$ is normal ring , then for any $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?
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Associated primes and their heights

Let $(S,m)$ be a commutative Gorenstein local ring, $I$ an ideal of $S$ such that $\operatorname{ht} I=t$, and $R=S/I$. Let $a \in m$ be an $R$-regular element such that for any prime ideal ...
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42 views

Find the projective closure of the ideal $I=\langle y-x^2,z-x^3\rangle$

When I looked at this example, my first instinct was to homogenize only the generators of $I=\langle f_1 := y-x^2,f_2:=z-x^3\rangle$ in a new variable $w$. But then, I realized that I would miss some ...
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44 views

Intersection and Sum of Polynomial Ideals from different rings

It is well known that intersection and sum of polynomial ideals from the same ring are lattice operations. I wonder if this is still true for ideals from different rings (over the same field). ...
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Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
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If $R\to S$ is a ring homorphism with $J$ an ideal of $S$. Show that the preimage of $J$ is an ideal of $R$.

Let $\alpha\colon R\to S$ be a ring homomorphism. Let $J$ be an ideal of $S$, and define the preimage of $J$ by $\alpha^{−1}(J)=\{r\in R\mid \alpha(r)\in J\}$. Show that $\alpha^{-1}(J)$ is an ideal ...
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Should $0$ be considered a prime?

Typically, a prime is defined as follows: $p$ is prime iff $(p \mid xy \implies p \mid x$ or $p \mid y)$ and $p$ is not a unit or zero. But for ideals, we say the zero ideal is prime. There is a ...
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Mistake in the proof that a domain is flat as a module over any subring

Where is the mistake in the following argument? I feel that there has to be one, for example by the very existence of this article. Let $R$ be an integral domain and $S \subseteq R$ be a subring ...
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Example showing that the product of ideals must be the span of the commutators

I'm trying to find an example showing why, in a Lie algebra, we can't just define the product of two ideals $I$ and $J$ to be the elements of the form $[x,y]$ where $x \in I, \; y \in J$. I imagine ...
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Maximal ideal in commutative ring

Let $R$ be a Dedekind domain, $\mathfrak{m}$ be a maximal ideal in $R[x]$ is of the form $\mathfrak{m} = (\mathfrak{p},f(x))$ where $\mathfrak{p}$ is a maximal ideal in $R$, and $f$ is a polynomial in ...
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Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
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69 views

A question about the consequence of Prime Avoidance.

I have found the following statement: Let $R$ be a Noetherian ring and $x$ is a non-zero divisor of $R$. Let $P$ be a prime ideal associated to $xR$. Then by Prime Avoidance there exists a non-zero ...
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Ideals (one-sided ideals) of $n×n$ upper triangular matrices

Is there any characterization of ideals (one-sided ideals) of $n\times n$ upper triangular matrices? I have just seen in monthly journal about $2 \times 2$ matrices in the below article Left and Right ...
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Finding reduced quadratic numbers and principal ideals

Hello :) I want to compute alle reduced quadratic numbers with discriminat $65$. We call a number $\gamma$ reduced if $\gamma>0$ and $-1<\gamma'<0$. We are working in quadratic field ...
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163 views

Find all ideals with given norm

I'm working through a final exam from 2 years ago. First task was to find the ideal class group of $\Bbb{Q}(\sqrt{-73})$. That is not the difficult work. I can give the 4 representants of the group by ...
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61 views

Subring of Z[x] with ACCP property

I have the subring of $\mathbb Z[x]$ with $f(1/2)$ always an integer. I have to check if it has ACCP (Ascending Chain Condition on Principal ideals) property or not. Any help would be greatly ...
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142 views

What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,…]$

What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,...]$? (this is the ring of polynomials over the reals with countably infinite many indeteminates). My attempt: I think taking the principal ...
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Simple $R$-module where $R$ is a semisimple ring. Possible small improvement of a proof.

Reading through the proof of the following theorem (in Introduction to Group Rings, by Milies and Sehgal) Let $L$ be a minimal left ideal of a semisimple ring $R$ and let $M$ be a simple ...
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Ideals of $\Bbb Z/p^2q\Bbb Z$

Let $p,q$ be distinct primes. Then $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct prime ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 2 distinct prime ...
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Ideals - A Geometric Interpretation?

The standard way to define an ideal is as follows: $I$ is an ideal if it satisfies the following conditions: $(I,+)$ is a subgroup of $(R,+)$ $\forall x \in I$, $\forall r \in R :\quad ...
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38 views

Examples of PIDs and prime ideals

(a) Give a specific example of a PID with exactly two prime ideals. Give a brief proof of your answer. (b) Give an specific example of a PID with infinitely many prime ideals. Give a brief proof of ...
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19 views

Descending chain condition on a finite dimension algebra

In a proof I'm reading, the author says "As $A$ is finite dimensional, a descending chain of left ideals must stabilize." The context is that $A$ is a finite dimensional simple $k$-algebra i.e. it ...
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Question about Principal Ideals

I'm just learning basic ring theory and had a question about the definition of a principal ideal. For a commutative ring $R$ with unity, Fraleigh defines the principal ideal generated by $a\in R$ as ...
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65 views

is there a criterion that says whether an ideal is radical or not?

Let $R=k[x,y,z]$. Is there a criterion that says whether an ideal of $R$ is radical or not? thanks
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51 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where p is prime integer and f is primitive integer polynomial that is irreducible ...
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93 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...