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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Clarification on notation in Siegfried Bosch's Commutative Algebra book about primary decomposition of ideals.

I'm reading through Siegfried Bosch's Commutative Algebra book, and I'm confused on his notation in one his proofs. He uses this notation a lot, so I think I should I understand it. The notation first ...
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Prove that $I_1^m\cap I_2^m \cap \dots \cap I_r^m=I_1^m\cdots I_r^m$, where $I_1,…,I_r$ are ideals in $k[x_1,…,x_n]$ and are comaximal.

This is an exercise from Ideals, Varieties and Algorithms by Cox, etc. If $I_i$ and $J_i=\cap_{j\ne i}I_j$ are comaximal for all $i$, where $I_1,...,I_r$ are ideals in $k[x_1,...,x_n]$, prove that ...
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When $f(I)S=S$ for each ideal $I$ of $R$?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). Question 1. ...
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An alternate definition of ideals

A filter on a poset if by definition its nonempty subset $F$ such that it does not contain the least element and $A, B \in F \Leftrightarrow \exists Z \in F : (Z \le A \wedge Z\le B)$ for every ...
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Localization of ideals at all primes

Let $R$ be a commutative ring with $1$ and $I$, $J$ ideals in $R$. For a prime ideal $P$, let $I_P=(R-P)^{-1}I$ be the localization of $I$ at $P$. Question: If $I_P=J_P$ for all prime ideals ...
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In a reduced ring the set of zero divisors equals the union of minimal prime ideals.

If $R$ is a reduced commutative ring with identity, why is the set $Z$ of zero divisors the union of minimal prime ideals? I know that $Z$ is a union of associated primes, and that the ...
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Express the ideal $(6) \subset\mathbb Z[\sqrt {-5}]$ as a product of prime ideals.

Express the ideal $(6) \subset\mathbb Z[\sqrt {-5}]$ as a product of prime ideals. I know I can write $(6)=(2)(3)=(1+\sqrt {-5})(1-\sqrt {-5})$. But I guess these factors might not be prime. ...
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If the intersection of ideals $I_{1},\ldots,I_{n}$ is contained in a prime ideal $P$, then one of them is contained in $P$

Let $A$ be a commutative ring and $I_{1},\ldots, I_{n}$ and $P$ ideals in $A$ with $P$ prime so that $\cap_{i=1} ^{n} I_{i} \subset P $. Show that there's an $i_0 \in \{1,...,n \}$ so that $I_{i_0} ...
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Height and minimal number of generators of an ideal.

Can anyone could give me a reference in a book about the proof of the following Let $I$ be an ideal of a ring. We denote with $\operatorname{ht}(I)$ the height of $I$, and by $\mu(I)$ the minimal ...
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If $\mathcal{L}$ is a minimal left ideal of an algebra $\mathcal{A}$, then $\mathcal{A}l=\mathcal{L}$ for all $l\in \mathcal{L}$?

I am going through Hassani's 'Mathematical Physics': He defines an algebra $\mathcal{A}$ as a vector space together with a multiplication map $\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$. He ...
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Showing there are at least one and only finitely many maximal ideals containing the extension of a maximal ideal [closed]

Let $F$ be a field and $M$ a maximal ideal of $F[x_1, x_2, ..., x_n]$. Let $K$ be an algebraic closure of $F$. Show that $M$ is contained in at least one and in only finitely many maximal ideals of ...
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generating element of $I = \{p \in \mathbb{Q}[X]: p(0)=0, p'(0)=0\}$

I have the ideal $I = \{p \in \mathbb{Q}[X]: p(0)=0, p'(0)=0\}$. I have verified that it is an ideal by multipyling an arbitrary element of the ideal with an arbitrary element of $\mathbb{Q}[X]$ and ...
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Prime ideal in $\mathbb Z[\sqrt{10}]$

I am trying to solve this exercise: Prove that $\langle 2,\sqrt{10} \rangle$ is a prime ideal in $\mathbb Z[\sqrt{10}]$. I could do the following: I pick an element of the form $zw \in \langle ...
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Can we make sense, in general, of taking a quotient by multiple ideals?

I feel that this is a rather silly question, stemming from a fundamental misunderstanding of quotients, but I'm not quite able to make it precise. My question is: given two ideals ...
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Left ideals in a subring of $M_2(R)$

Let $R$ be a 2-dimensional complete regular local ring $R$ over an algebraically closed field $k$, that is $R\cong k[[x,y]]$. Now look at the the following subring $A$ of $M_2(R)$: $A=\begin{pmatrix} ...
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Using Koszul complex [closed]

Let $A$ be a Noetherian local ring of dimension $t$ with maximal ideal $\mathfrak{m}$. If $J\subset A$ is an $\mathfrak{m}$-primary ideal then we have the following complex for $n\in \Bbb N$: ...
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show set is prime ideal

Let I = { (a,0): a E Z} A)show that I is a prime ideal of Z X Z B) by considering (ZXZ)/I , or otherwise , determine whether I is a maximal ideal of ZXZ. (0,0) is in I so I is non-empty let (a,0) ...
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Principal ideals containing an ideal in a Noetherian integral domain

Let $R$ be a Noetherian integral domain and $I$ a nonzero ideal consisting only of zero divisors on $R/(x)$, where $x$ is a nonzero element of $I$. Could we always find an element $y\notin (x)$ such ...
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Prove $I$ is non-principal ideal of $\mathbb{Z}[x]$? [duplicate]

I'm new to algebra and got stuck with concept of ideals. The question is to prove that $$I = \left\{ {{a_0} + {a_1}x + \cdots + {a_n}{x^n} \mid {a_i} \in \mathbb{Z},{a_0} \in 2\mathbb{Z}} ...
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Length of a ring? Lenth of a (right or left) ideal

I have seen the concept of length being applied to rings. What is exactly mean by that? What does length mean in a statement like "the composition length of RR is 2, but the composition length of RR ...
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Tensor products of ideals

Consider the class of complex algebras where the tensor products are over complex numbers. Given a complex algebra $A$ and a left ideal $L$ of $A$ generated by $n$ elements. Is $L^{\otimes n}$ ...
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Lasker-Noether Theorem and Kummer-Dedekind

I would like to know about the relations between Ernst Kummer's invention of complex ideal numbers (and Dedekind's development of them into what is now called ideals) regarding the unique ...
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A question regarding one-sided right and left ideals (their calculation for matrix rings)

I have a question regarding the calculation of right and left ideals of a matrix ring. I understand the concept, or so I believe, but, when consulting the argument on this page ...
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If $I$ and $J$ are ideals in $R$, and $I$ is a subset of $J$, is $I$ also an ideal in $J$?

Pretty much what the title suggests. If $I \subset J$ are both ideals in a commutative ring $R$, is it true that $I$ is an ideal in $J$? My reasoning for this is that clearly for all $a,b\in I$, $a ...
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If I is an ideal of $R$ ($R$ a ring) prove that $M_{n}(R/I)$ is isomorphic to $M_{n}(R)/M_{n}(I)$

If $I$ is an ideal of $R$ ($R$ a ring) prove that $M_{n}(R/I)$ is isomorphic to $M_{n}(R)/M_{n}(I)$ I proved that $M_{n}(I)$ is an ideal of $M_{n}(R)$ but I don't know how to prove this. Thanks for ...
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Distributive law of ideals in $\mathbf Z$ relating $\cap$ & $+$

Let $\mathfrak a, \mathfrak b$ and $\mathfrak c$ be ideals in $\mathbf Z$. Then show that $$ \mathfrak a \cap (\mathfrak b + \mathfrak c) = \mathfrak a \cap \mathfrak b + \mathfrak a \cap \mathfrak c ...
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Why is the dual of a filter an ideal?

Jech's set theory, (3rd edition) says that if $F$ is a filter on $S$ Let $I = \left\{ {S - X: X \in F}\right\}$ then $I$ is an ideal of $S$ (dual to $F$). However, let $X,Y \subset S$, $X \in I$ ...
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Is an irreducible ideal in $R$ irreducible in $R[x]$?

Let $R$ be a commutative Noetherian ring and $I\subset R$ an ideal that is irreducible in the sense that if $I = J_1 \cap J_2$, then $I=J_1$ or $I=J_2$. Is (the ideal generated by) $I$ irreducible in ...
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Why is $(XY-1)$ contained in $(X-a, Y-b)$ with $ab=1$?

This is probably a very trivial question, so I apologize in advance. Let $K$ be an algebraically closed field and $R=K[X,Y]$ the polynomial ring in two variables. I want to show that every ideal ...
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On the history of sigma-ideals

Could anyone provide me with some insight regarding the history of sigma-ideals, i.e., who coined them, first publications on the matter, main authors thereafter and so on? Thanks in advance.
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The opposite of the right ideal in the ring of 2x2 matrices?

Since every ring has an opposite, I would like to know: Which is the opposite of the rings of $n \times n$ matrices? More specifically, of the $2 \times 2$ matrices. Is there an opposite for the ...
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What is the class group of $\Bbb{Q}(\sqrt{-41})$?

What is the class group of $\Bbb{Q}(\sqrt{-41})$? I've found that it's generated by $P_2, P_3, P_5, P_7$ as per Dedekind's theorem, but I'm having a bit of trouble finding the relations between the ...
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Show the ideal $\langle x^2+1, 5\rangle$ isn't maximal

Show the ideal generated by $x^2+1$ and $5$ isn't maximal in $\mathbb{Z}[x]$. I thought of the following: An element of $\mathbb{Z}[x] /I$ is $f(x) + \langle x^2 +1 , 5 \rangle = ax+b + I$ ...
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Considering the prime ideals of $\Bbb Z$ to find the prime ideals of $\Bbb Z[x]$

Why can we consider the prime ideals of $\Bbb Z$ to determine the prime ideals of $\Bbb Z[x]$? There really isn't any work I can show here. Motivation is that it seems various papers consider a prime ...
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$\Bbb Z[x]/(p)\cong \Bbb F_p[x]$

How do I prove this? $$\Bbb Z[x]/(p)\cong \Bbb F_p[x]$$ This implies that $(p)$ must be equal to $p\Bbb Z$ where $p$ is prime. Why is the only prime element that makes this true $p$ where $p$ is ...
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Ideal $I\cap \Bbb Z\ne(0)$

What does this mean: $I$ is an ideal of $\Bbb Z[x]$. Let $I\cap \Bbb Z\ne(0)$ What does it mean to take the intersection of an ideal and the integers? All elements of the ideal that are integers? We ...
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Understanding a quotient ring of continuous functions

In trying to understand another questions answer(to a question I asked), I realized that my fundamental lack of knowledge was in regards to the following question: In terms of functions, what does ...
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Algebra and Maximal ideal.

I am trying to solve the following problem. If $ \mathcal{K}$ is a field and $a_1,a_2,\dots,a_n \in \mathcal{K}$. Prove that $(x_1-a_1,x_2-a_2,\dots,x_n-a_n)$ is a maximal ideal in ...
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Morita equivalence and right and left ideals of a Ring

I have been thinking a bit about Morita equivalence http://en.wikipedia.org/wiki/Morita_equivalence and I would like to know whether it also applies to subrings such as right or left ideals. And, if ...
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Number of ideals in a minimal irreducible decomposition

Assume $R$ is a local ring, $M\subseteq R$ is the maximal ideal, $I\subseteq R$ is an $M$-primary ideal and $I=\bigcap_{i=1}^n Q_i$ is a minimal irreducible decomposition of $I$ (i.e. $Q_i\subseteq R$ ...
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A basic isomorphism of modules (useful for Corollary 2.7 of Atiyah and MacDonald).

Let $M$ be an $A$-module, $N$ a submodule of $M$, $\mathfrak{a} \subseteq A$ an ideal such that $M = \mathfrak{a}M + N$. Then $\mathfrak{a}(M/N) = (\mathfrak{a}M+N)/N$ I am having troubles in ...
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Stably-free ideals are free?

In my class of algebraic topology, a friend of mine stated the following: If $R\ne 0$ is a commutative ring with unit and $I\subset R\oplus R$ is a submodule such that $(R\oplus R)/I\cong R$, ...
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Order of prime ideals over split primes in the class group.

Let $P$ be a prime ideal of $\mathcal O_K$ ($K$ a quadratic field) and let $P$ have norm $p$ where $p$ is a split prime. Is it possible for the ideal class $[P]$ to have order less than three? I feel ...
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Example of commutative ring that doesn't satisfy distribution of intersection over addition

I'm trying to find an example of commutative ring $R$ and ideals $\mathfrak a,\mathfrak b,\mathfrak c \in R$ such that $$\mathfrak a \cap (\mathfrak b + \mathfrak c) \neq \mathfrak a \cap ...
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Wrong proposition in “Atiyah and Macdonald”s book?!

In page 6 of "Introduction to commutative algebra" says that: $a \cap b = ab$ provided $a + b = (1)$ But i think it's not true,by considering $a = b = (2) \in \mathbb Z_6$
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How to workout what elements of a quotient ring look like?

I am trying to understand quotient rings. Firstly: $$\frac{\Bbb Z[x]}{\langle x-1\rangle}$$ The above I can understand in a fairly naive way. Since the ideal is generated by a degree one polynomial, ...
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Cardinality of quotient ring $\mathbb{Z_6}[X]/(2x+4)$

Let $R$ be the ring obtained by taking the quotient of $\mathbb{Z_6}[X]$ by principal ideal $(2x+4)$. Then 1) $R$ has infinite elements 2) $R$ is field 3) $5$ is unit in $R$ 4) $4$ is unit in $R$. ...
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Morita equivalence between right and left ideals of a ring

I would like to know whether Morita equivalence is a useful tool when dealing with right and left ideals of a ring. If so, could someone illustrate it on the example of $2\times 2$ matrices? Thanks
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Is there a classification of ideals of $\mathcal O_K$ ($K$ quadratic) over ramified and split primes depending on $d \pmod 4$?

I am unsure if the following argument is correct. I have not seen something like this in my course, so I'm a bit skeptical, since this seems like a very simple way of computing norms of ideals. If ...
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Writing the ideal $m=\langle X, Y \rangle$ in $R=k[X, Y]$ as a countable union of prime ideals

Here's a problem (Exercise 3.21) from "A Term in Commutative Algebra" by Altman & Kleiman: Let $k$ be a field, and $R=k[X, Y]$ be polynomial ring in two variables. Let $\mathfrak{m}=\langle ...