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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3answers
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Is $\lhd$ common notation for “is an ideal of”?
This question is because of this comment. I would like to know if I should refrain from using $\lhd$ for "is an ideal of" in ring-theoretic questions. Is it common enough, or should I explain what it ...
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0answers
33 views
the leading term of a module
I'm reading CLO and I have a question about the following Prop:
Let I be an ideal in a polynomial ring $k[x_1,\ldots,x_n]$. Then $k[x_1,\ldots,x_n]/I$ is isomorphic as a $k$-vector space to $S = ...
4
votes
5answers
345 views
If the localization of a ring $R$ at every prime ideal is an integral domain, must $R$ be an integral domain?
Let $R$ be a commutative ring. Suppose that for every prime ideal $p$ of $R$, the localized ring $R_p$ is an integral domain. Must $R$ be a integral domain?
I was trying to think of counter-examples, ...
2
votes
1answer
62 views
Hilbert function and isom of varieties
I'm just curious about a certain concept:
If two ideals $I$ and $J$ in a polynomial ring $R$ have the same Hilbert function (note: I'm not talking about the Hilb polynomial), then are their supports ...
2
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0answers
50 views
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 ...
3
votes
1answer
133 views
Generating set for sum of two ideals
Suppose there are two ideals $I,J \in \mathbb{C}[x_1,\dots,x_k]$ and two sets of generating polynomials $\langle f_1, \dots, f_s\rangle$, $\langle g_1, \dots, g_t\rangle$. Now I want to describe $I + ...
1
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2answers
54 views
Confused about principal rings
A ring $A$ is principal if every ideal in it is of the form $Ax$. $\mathbb{Z}$ is said to be a principal ideal, but it seems to me that a set $I = \{z \in \mathbb{Z} : |z| > n\} \cup \{0\}$ for ...
2
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2answers
182 views
What does the word “ideal” mean in this context?
I'm confused about a terminology.
In Frank W. Warner's book Foundations of Differentiable Manifolds and Lie Groups, it says on page 12
Let $F_m$, a subset of $\bar{F_m}$ (the set of germs at ...
2
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1answer
71 views
Technical Lemma for Central Simple Algebras?
There is a technical lemma on slide 7 of these slides, but no proof.
Is there a simple proof I can read before moving on? The lemma itself says
Let $S$ be a central simple $k$-algebra and let ...
4
votes
2answers
98 views
Does totally flat commutative ring imply all ideals are idempotent?
From reading Atiyah and MacDonald, I know of the result that a absolutely flat commutative ring has all principal ideals idempotent.
Reading around on math reference, I think that if a commutative ...
1
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2answers
177 views
Followup to “Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$”
In this post: Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$ a nice example was given of a non-distributive ring. The lattice of ideals turned out to be the Diamond lattice $M_3$ with ...
0
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1answer
236 views
Nilpotent elements in a quotient ring.
Let R be a ring. Let I be an ideal of R. If R/I doesn't have nonzero nilpotent element, every nilpotent element in R is contained in I. Then, if I contains every nilpotent element in R, there is no ...
5
votes
2answers
217 views
(Minimal?) Polynomials using the Nullstellensatz
I'm struggling with an exercise that was asked in class:
Let $\alpha = \sqrt[3]{3} + \sqrt{7}\sqrt[4]{2}.$ Show that there is a polynomial $p$ in
the ideal $I=\left<a^3 - 3, b^2 - 7, c^4-2, ...
0
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1answer
412 views
Commutative Ring: Nilpotent elements closed under addition? [duplicate]
Possible Duplicate:
The set of all nilpotent element is an ideal of R
Given a commutative ring $R$ and two nilpotent elements $r$, $s$ there exists an $n \in \mathbb{N}$ such that
$$ ...
2
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1answer
96 views
Why does uniqueness of prime ideal factorization imply that “contains means divides”?
I previously posted about this here: comaximality of ideals in a commutative ring with unit
Sadly, having an unregistered account at the time, I can't edit that post. I will say thanks to Arturo ...
0
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2answers
155 views
What is $\langle x + y\rangle + \langle x - y \rangle$?
Let $R$ be a ring. Let $I = \langle x + y \rangle$, $J = \langle x - y \rangle$ be ideals of $R[x,y]$.
What's $I + J$ in this case? By definition $I + J = \{ i + j \mid i \in I, j \in J \}$. My first ...
1
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1answer
197 views
Definition of Ideals generated by a set
I'm struggling to understand the definition of ideals in ring homomorphisms generated by a set.
If $R$ is commutative and has a $1$, then Ideal of $R$ generated by a subset $A$ of $R$:
$$⟨ A ⟩ = ...
7
votes
2answers
124 views
Countable rings
Suppose we are given a countable unital ring $R$ with uncountably many distinct right ideals. Does it follow from this that $R$ has uncountably many maximal right ideals?
1
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1answer
181 views
Rings such that $A[x]$ is a principal ideal domain
Let $A$ be a commutative ring.
Then the following assertions are equivalent.
$A$ is a field;
$A[x]$ is a Euclidean domain;
$A[x]$ is a principal ideal domain;
$A[x]$ is a unique factorization ...
3
votes
2answers
215 views
Maximal Ideals of direct products
Maximal Ideals of $R\times S$ are either of the form $A \times S$, where $A$ is maximal in $R$, or of the form $R\times B$, where $B$ is maximal in $S$.
I started by assuming $U$ is maximal in $R ...
3
votes
0answers
106 views
Can the ideal $(X_1, X_2, \dots, X_n) $ be generated by fewer polynomials over the field $K[X_1, X_2, \dots, X_n]$?
My algebra teacher asked whether the ideal $(X_1, X_2, \dots, X_n) $ can be generated by fewer polynomials over the field $K[X_1, X_2, \dots, X_n]$.
My intuition tells me that it can't, so I tried to ...
6
votes
1answer
281 views
How to show this ideal is not principal
I have been brushing up on cubic number fields. Specifically, let $s$ be a root of the polynomial $x^3 + x^2 + 3x + 17$, and consider $K = \mathbb{Q}(s)$; we have $\mathcal{O}_K = \mathbb{Z}[s]$, and
...
3
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1answer
335 views
$I+J=R$, where $R$ is a commutative rng, prove that $IJ=I\cap J$.
So I basically have to prove what is on the title. Given $R$ a commutative rng (a ring that might not contain a $1$), with the property that $I+J=R$, (where $I$ and $J$ are ideals) we have to prove ...
3
votes
1answer
118 views
Minimal generating sets for homogeneous polynomial ideal in two variables
This question is (somehow) related to System of generator of a homogenous ideal
Let $A$ be the ring $A={\mathbb R}[X,Y]$, and let $m \geq 1$. Let
$$
{\cal S}_m=\lbrace X^m, X^{m-1}Y,X^{m-2}Y^2, ...
5
votes
0answers
155 views
A ring that has exactly 7 left ideals (T. Y. Lam)
Exercise 3.25 in Lam's First Course states:
Let $R$ be a ring that has exactly seven nonzero left ideals. Prove that one of them is an ideal (i.e. left and right) and provide an example of such a ...
4
votes
2answers
251 views
How to find the nilpotent elements of $\mathbb{Z}/(\prod p_i^{n_i})$?
I've been following MIT's old opencourseware class on commutative algebra. For one problem, I want to find the nilpotent and idempotent elements of $\mathbb{Z}/(n)$, where $n=\prod p_i^{n_i}$.
I know ...
2
votes
4answers
274 views
Why are powers of coprime ideals are coprime? [duplicate]
Possible Duplicate:
Comaximal ideals in a commutative ring
On this webpage http://www.imsc.res.in/~kapil/geometry/caag/finite.html it's a stated fact that if $I$ and $J$ are coprime ideals ...
4
votes
2answers
116 views
Why is the kernel of $k[x_1,\dots,x_n]\to k$ a maximal ideal?
In Reid's Undergraduate Commutative Algebra, $k$ a field and a point $P=(a_1,\dots,a_n)\in k^n$ determine a homomorphism on the the polynomial ring of functions $k[x_1,\dots,x_n]\to k$ by $g\mapsto ...
4
votes
2answers
238 views
Is this using the first isomorphism theorem for rings?
Let $F$ be a field and $f(x) = x - 1$ and $g(x) = x^2 - 1$.
1) Show that $F[x]/(f(x)) \cong F$
2) Is ideal $(g(x))$ maximal? Explain your answer.
** I have a feeling that this uses the first ...
5
votes
1answer
143 views
In $K[X,Y]$, is the power of any prime also primary?
I've recently been reading about primary decomposition, and was browsing the questions here.
From this, I know that it is not true that every primary ideal is the power of a prime ideal.
I'm curious ...
4
votes
1answer
202 views
Length of maximal chain of prime ideals equals transcendence degree of fraction field?
I've been reading some commutative algebra, but have been struggling with this idea for a while.
Let $k$ be a field, and let $A=k[x_1,\dots,x_n]$ be a finitely generated integral domain, such that ...
3
votes
1answer
159 views
Ideals in the ring of endomorphisms of a vector space of uncountably infinite dimension.
I know that if $V$ is a vector space over a field $k,$ then
$\operatorname{End}(V)$ has no non-trivial ideals if $\dim V<\infty;$
$\operatorname{End}(V)$ has exactly one non-trivial ideal if ...
2
votes
1answer
324 views
When is a product of two ideals strictly included in their intersection?
Let $I,J$ two ideals in a ring $R$. The product of ideals $IJ$ is included in $I \cap J$. For example we have equality in $\mathbb{Z}$ if generators have no common nontrival factors, in a ring $R$ ...
4
votes
1answer
177 views
Product of a principal proper ideal by itself
Let $P$ be a principal proper ideal in an integral domain.
Is it $P^2 \subset P$ in general? If yes, how to prove it?
For example, if you look at the ideal $(3)=3\mathbb{Z}$ in $\mathbb{Z}$, it ...
0
votes
1answer
278 views
Notation about sums and direct sums of ideals
Let $A_1,A_2,\dots,A_n$ be a family of (right) ideals. $A_1+A_2+\cdots +A_n$ is the sum of these ideals (it is the smallest ideal containing the $A_i$'s). Another way to denote this sum is ...
0
votes
2answers
201 views
Ideals of Subrings
Show by example that not every ideal of a subring $S$ of a ring $R$ need be of the form $I\cap S$ for some ideal $I$ of $R$
If $I$ is an ideal of $R$ and $S$ is a subring of $R$, then it can be ...
1
vote
2answers
211 views
Show that $I$ is an ideal of $R$
Let $R$ be a commutative ring and let $a\in R$.
Show that $I=\{x\in R\mid ax=0\}$ is an ideal.
For all $b \in R$, $$bI=b\{x\in R\mid ax=0\}=\{bx\in R\mid a(bx)=0\} =\{xb\in R\mid ...
5
votes
1answer
104 views
Three maximal ideals lying over $3\mathbb{Z}$?
A few weeks ago I asked a question about finding the number of maximal ideals lying above $3\mathbb{Z}$ in $B$, where $B$ is the integral closure of $\mathbb{Z}$ in a splitting extension ...
2
votes
1answer
198 views
Find all the prime ideals of $\{\frac{a}{b}| a \in \mathbb{Z}, b \in \mathbb{N}_0 \text{ odd}\}$
For an exercise in my book I have to find all the prime ideals of $$R = \left.\left\{\frac{a}{b}\;\right|\; a \in \mathbb{Z}, b \in \mathbb{N}_0 \text{ odd}\right\}\leq (\mathbb{Q},+,\cdot)$$
I ...
1
vote
2answers
139 views
Find a left ideal of $\mathbb{H}[X]$ that is a maximal, but not a ideal.
Find a left ideal of $\mathbb{H}[X]$ that is a maximal, but not a ideal.
$\mathbb{H}[X]$ is just the polynomial extension.
$$\mathbb{H}=\{a+bI+cJ+dK \mid a,b,c,d \in \mathbb{R} \} .$$
...
4
votes
1answer
309 views
Are distinct prime ideals in a ring always coprime? If not, then when are they?
Essentially as the title suggests - in some commutative ring $K$ (with 0,1), if we have 2 distinct proper prime ideals $\mathfrak{p}_1 \neq \mathfrak{p}_2$, is it necessarily the case (or if not, when ...
2
votes
1answer
188 views
Ideal in the product of two rings:
$R$ and $S$
are two ring, let $J$
ideal in $R\times S$
then there are $I_{1}$
ideal of $R$
and $I_{2}$
ideal of $S$
such that $J=I_{1}\times I_{2}$
For me is abvious why $\left\{ r\in ...
-2
votes
1answer
534 views
Prove that R is a field ↔ The only ideals in R are R(0) and (0). [duplicate]
Possible Duplicate:
Rings and ideals
The question is the title, any help would be greatly appreciated!
2
votes
2answers
87 views
Set of associated primes of direct sum
Let $M$ be a module over the ring $R$.
Let $\operatorname{Ass}(M)$ be the set of annihilator ideals $\operatorname{Ann}(x)$, which are prime, so
$$\operatorname{Ass}(M) = \{\operatorname{Ann}(x) ...
3
votes
1answer
610 views
If $I$ is a maximal ideal of $R$, why is $R/I$ a field?
If $I$ is a maximal ideal of $R$, why is $R/I$ a field?
I'm trying to use the fact that $I$ is maximal to show that $R/I$ only have ideals $\{0\}$ and $R/I$. Can anyone help me with this method. Many ...
2
votes
1answer
135 views
Extending an ideal of a polynomial ring to a polynomial ring with more indeterminates. Is it a tensor product?
Let $\mathbb{k}$ be a field, let $S'=\mathbb{k}[x_1,x_2,\dots,x_m]$, and let $I'\subseteq S'$ be an ideal.
For some $n>m$, let $$S=\mathbb{k}[x_1,x_2,\dots,x_n]\ \ \ ...
5
votes
2answers
231 views
Are ideals in rings and lattices related?
There are (at least) two notions of ideals:
An ideal in a ring is a set closed under addition and multiplication by arbitrary element.
An ideal in a lattice is a set closed under taking smaller ...
2
votes
2answers
297 views
Height and minimal number of generators of an ideal
How can I determine the height and the least number of generators of the ideal
$ I=(xz-y^2,x^3-yz,z^2-x^2y) \subset K[x,y,z] $?
I tried to calculate the dimension of the vector space ...
1
vote
0answers
175 views
Free modules and Polynomial rings
Given a field $K,$ we have the polynomial ring $K[x,y]$ in $2$ variables, which is also a left module (over itself). How can we prove that the ideal $(x,y)$ is not a free module?
8
votes
2answers
521 views
Methods to check if an ideal of a polynomial ring is prime or at least radical
I am looking for methods to check whether a given ideal in $K[x_0,\dots,x_n]$ is prime. I mean something you can effectively use in some concrete non-trivial example.
To be more explicit, I am working ...
