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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A basic isomorphism of modules (useful for Corollary 2.7 of Atiyah MacDonald).

Let $M$ 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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Applications of $\mathbb{Z}/n\mathbb{Z}$ [on hold]

I would like someone to proof me this claim and give me its applications in mathematics if it's not a convention. Claim: for all positive integers $n$, the ring $\mathbb{Z}/n\mathbb{Z}$ is domain if ...
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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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27 views

Cardinality of Quotient ring

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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How to visualize one-side ideals of a ring?

Could anyone provide me with some insight as to how to represent diagrammatically (in a graphical o visual way) left and right ideals of a ring (in those cases in which they are one-sided ideals)? ...
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Prime ideal in Dedekind ring is finitely generated

Let $R$ be a Dedekind ring, which means integral domain, integrally closed, Noetherian, which means that given any chain of ideals in $R$: $$I_1\subseteq\cdots \subseteq I_{k-1}\subseteq ...
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Who coined ideals in Set Theory?

One of the meanings of the word "ideal" in maths refers to Set Theory. Even though handbooks say that concept can be translated to Order Theory or to Algebra effortelssly, I am interested in: 1) ...
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In which way are sigma ideals a special case of ideals?

The article on sigma-ideals in wikipedia claims they are a special kind of ideals: http://en.wikipedia.org/wiki/Sigma-ideal But, unfortunately, no explanation to that regard is offered (not at least ...
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Why are principal fractional ideal also fractional ideals?

I don't understand the following: Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ other than $\{0\}$, for which a $0\neq r\in R$ exists, so that ...
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Relation of ideals in probability with other kinds of ideals?

It seems that there are at least 5 kinds of ideals in maths: Ideals in number theory (Kummer, Dedekind) Ideals in abstract algebra (Dedekind, Noether), as kernels of homomorphisms Ideals in order ...
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How do I compute the norm of a non-principal ideal of the ring of integers of a quadratic field without using ''large'' results

I am trying to compute the norm of the ideal $I=(7, 1+\sqrt{15}) \trianglelefteq \mathbb Z[\sqrt{15}],$ the ring of integers of $\mathbb Q[\sqrt{15}].$ I knew $I^2$ would be principal, as $I\bar ...
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prove $(m) \subset (n)$ iif $n$ divides $m$

For non-zero integers $m$ and $n$, prove $(m) \subset (n)$ iif $n$ divides $m$, where $(n)$ is the principal ideal. My attempt is following. For non-zero integers $m$ and $n$, assume that $(m) ...
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Relation between Noether's one-sided ideals and Polish notation?

Given the definitions of one-sided ideals (right ideals; left ideals) bu Emmy Noether, as referred in this answer Noether's definition of right and left ideals?, I would like to raise the ...
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23 views

Origin of ideals in order theory

I am trying to clarify in my head the different meanings of "ideals" in mathematics. We have ideals in Number Theory, as in Dedekind (derived from 'Ideal Complex Numbers' in Kummer), in Abstract ...
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A question on Join homomorphism and Ideals

On page 287 of the book Mathematical Methods in Linguistcs, by Barbara Partee, Alice Ter Meulen and Robert E. Wall (Dordrecht, Kluwer Academic Press, 1993), I find the following theorem, which they ...
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+50

What does the ideal norm of matrix elements really mean?

Say we have a number field $K$ (specifically, an imaginary quadratic field) and a $2\times2$ matrix $\sigma=\pmatrix{a&c\\b&d}$ with elements $a,b,c,d\in\mathcal O_k$, the ring of integers of ...
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Visual representation of (right or left) ideals of a ring?

I would like to know about the canonical visual representation for ideals in a ring. Particularly, for the two kinds of one-sided ideals,that is, right ideals and left ideals. Is it possible to use ...
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Decision between right and left ideals of a ring?

Let us suppose whe find a phenomenon (in nature, social sciences, whatever) for which we believe (or some author has stated) it is possible a formalization or modeling in terms of ideal of a ring. Let ...
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Class group of $\mathbb Q(\sqrt{-55})$ and finding representatives for ideal classes

My first step in computing the class group of $\mathbb Q(\sqrt{-55})$ was to compute the Minkowski bound. Initially, I said $\lambda(-55)=2\sqrt{-55}/\pi<2(8)/3<6$ and I went the normal way of ...
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21 views

Relation between ideal numbers and ideals of a ring?

I would like to know whether the ideal numbers of Kummer (or the ideals of Dedekind for that matter) are closely related to the concept of ideal (right ideal, left ideal, two-sided ideal and so ...
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All polynomial parametric curves in $k^2$ are contained in affine algebraic varieties

I have started working through the textbook Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea and I am stuck on one part of an introductory question. The question begins by getting one to ...
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48 views

Is the minimal number of generators of an ideal the rank of the ideal as a free $\mathbb Z$-module?

In an algebraic number theory course, my lecturer said that any ideal of $\mathcal O_K$, where $K$ is a quadratic number field, is generated by at most two elements. I am wondering why this is. When ...
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83 views

Question about ideals of a ring: $I\cdot J=I \implies J=I$?

Doing exercises, this question came to my mind. Is it true that if $I$ and $J$ are proper and nonzero ideals of a ring $R$, $$I\cdot J=I \implies J=I?$$ And $$I\cdot J=I \iff J\subseteq I?$$
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Residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$

What are the residue fields of the closed points in $Spec(\mathbb{R}[X,Y])$? After finding the maximal ideals of $\mathbb{R}[X,Y]$, which are of the form: $\langle X-a,Y-b \rangle$ with $a,b ...
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Problem about ideals of the localization of a ring

I'm having problems on doing the section (ii) of this exercise. Let $R$ be a domain. Let $P$ be a prime ideal of $R$. (i) Prove that $S:=R\setminus P$ is a multiplicatively closed system with no ...
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Direct sum of non-zero ideals over an integral domain

Let $R$ be an integral domain. Let $I$ and $J$ be non-zero ideals of $R$. Is this statement always true: $$R\oplus(I\cap J)\cong I\oplus J\ ?$$ I regarded the short exact sequence $0\to I\cap ...
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$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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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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1answer
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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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+100

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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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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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. ...