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, ...

learn more… | top users | synonyms

0
votes
3answers
48 views

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 ...
1
vote
1answer
38 views

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 ...
3
votes
0answers
46 views

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} ...
2
votes
1answer
53 views

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 $n\in \Bbb N$: ...
1
vote
1answer
37 views

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) ...
1
vote
1answer
25 views

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 ...
3
votes
3answers
85 views

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}} ...
0
votes
1answer
23 views

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 ...
1
vote
0answers
32 views

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 $L^{\otimes n}$ ...
1
vote
0answers
30 views

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 ...
0
votes
0answers
18 views

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 ...
3
votes
3answers
44 views

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 ...
0
votes
1answer
24 views

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 ...
2
votes
2answers
31 views

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 ...
2
votes
2answers
45 views

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 $X,Y \subset S$, $X \in I$ ...
0
votes
0answers
71 views

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 ...
1
vote
3answers
56 views

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 ...
0
votes
0answers
15 views

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.
0
votes
0answers
39 views

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 ...
3
votes
2answers
70 views

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 ...
1
vote
2answers
45 views

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 $f(x) + \langle x^2 +1 , 5 \rangle = ax+b + I$ ...
1
vote
0answers
36 views

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 ...
2
votes
1answer
46 views

$\Bbb Z[x]/(p)\cong \Bbb F_p[x]$

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 ...
0
votes
2answers
37 views

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 ...
0
votes
0answers
37 views

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 ...
0
votes
2answers
55 views

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 ...
0
votes
1answer
28 views

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 ...
2
votes
1answer
78 views

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. $Q_i\subseteq R$ ...
1
vote
2answers
55 views

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 ...
4
votes
4answers
140 views

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 $(R\oplus R)/I\cong R$, ...
3
votes
0answers
44 views

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 ...
2
votes
3answers
117 views

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 ...
0
votes
1answer
58 views

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 $a = b = (2) \in \mathbb Z_6$
2
votes
2answers
73 views

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, ...
1
vote
1answer
75 views

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 $R$. ...
0
votes
0answers
10 views

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
2
votes
0answers
63 views

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 ...
3
votes
2answers
94 views

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 ...
0
votes
0answers
14 views

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)? ...
2
votes
1answer
35 views

How to show the existence of an ideal in the ring of Gaussian integers that satisfy the following?

How to show that If $p$ is a prime and $p\equiv1\bmod4$ then there exists an ideal of the $R=\mathbb{Z}[i]$, the ring of Gaussian integers, such that $R/I$ is isomorphic to ...
0
votes
1answer
36 views

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 ...
0
votes
0answers
50 views

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) ...
0
votes
1answer
50 views

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 ...
1
vote
0answers
25 views

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 ...
2
votes
0answers
37 views

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 ...
0
votes
2answers
41 views

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 ...
1
vote
2answers
55 views

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) ...
0
votes
0answers
14 views

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 ...
2
votes
1answer
26 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 ...
2
votes
2answers
48 views

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 ...