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, ...
3
votes
2answers
243 views
Does every prime ideal in a ring arise as kernel of a homomorphism into $\mathbb{Z}$?
Let $R$ be a commutative ring. Clearly the kernel of $h$ is a prime ideal whenever $h : R \rightarrow ...
1
vote
2answers
384 views
proof of chinese remainder theorem for ring
Let $R$ be a ring(not necessary have "1") and let $I,J$ be ideals of $R$ such that $I+J=R$. I want to prove that there is a $x\in R$ such that
$$x\equiv r ({\rm mod} I) \quad x \equiv s ({\rm mod} J) ...
4
votes
1answer
95 views
Idempotent and Hermitian vectors in Group Algebra
Let $C$ be the field of complex number and $G$ a finite group, then define $C[G]$ be a vector space over $C$, with elemnts of $G$ as the basis. Then any element in $C[G]$ can be written as $\sum_{g ...
3
votes
2answers
103 views
What letter should I use to denote an ideal?
In commutative algebra, there seem to be two rather different notational conventions for ideals: either $I,J, \dots$ or $\mathfrak{a}, \mathfrak{b}, \dots$.
By itself, it is hardly surprising - after ...
0
votes
1answer
203 views
multiple choice question for compact support functions.
Let $C(\mathbb R)$ denote the ring of all continuous real-valued functions on $\mathbb R$, with the operations of pointwise addition and pointwise multiplication. Which of the following form an ideal ...
1
vote
1answer
85 views
Commutants of commutative algebras
Let $W$ be a unital algebra and let $V$ be its maximal abelian subalgebra. Must the commutant $V^\prime$ of $V$ be commutative?
2
votes
1answer
174 views
Quotient ring of a polynomial ideal with two variables
Given an ideal $I = \langle x-y,y^3+y+1 \rangle \subset \mathbb{C}[x,y]$ (this is a Gröbner basis w.r.t. degree-lexicographic order). I want to write $\mathbb{C}[x,y]/I$ as a $\mathbb{C}$-Basis and ...
2
votes
1answer
56 views
For which $m \in \mathbb N$ is the ideal $(m,x^2+y^2)$ prime in $\mathbb Z[x,y]$?
Let $m \in \mathbb N$. Find a necessary and sufficient condition for
$m$ such that the ideal $(m,x^2+y^2)$ is prime in $\mathbb Z[x,y]$.
I have to find for which $m$ the quotient ring is an ...
1
vote
1answer
168 views
Annihilator of a simple module
Let $R$ be a finitely generated commutative ring and $C$ an $R$-algebra ($C$ is not necessarily commutative). Assume that $C$ is a finitely generated $R$-module.
If $S$ is a simple $C$-module, then ...
13
votes
0answers
199 views
What is the algebraic structure of functions with fixed points?
So I just noticed that the set of functions with a fixed point
$$f(x_0)=x_0,$$
are closed under composition
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
9
votes
5answers
329 views
How does a Class group measure the failure of Unique factorization?
I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
3
votes
1answer
120 views
Why is $S/Z$ a domain for the ideal $Z=\{z\in S\mid za=0,\;\forall a\in R\}$ in $S$?
Suppose $R$ is a rng with no zero divisors, not necessarily commutative. I know $R$ can be embedded into a ring $S:=\mathbb{Z}\times R$ by identifying $r\in R$ with $(0,r)\in S$. The operations on $S$ ...
2
votes
1answer
72 views
A question on artinian semi-primitive rings
So the question is as follows:
Suppose $U$ is an ideal of artinian ring $R$, then show that there is an ideal $V$ such that $U+V=R$ and $U\cap V \subseteq J(R)$ .
Let me describe my approach. I took ...
2
votes
1answer
106 views
A problem on the Jacobson radical, from Isaacs Graduate Algebra
This is problem 14.10 from Isaacs Graduate Algebra.
Let $U$ and $V$ be ideals of a ring $R$ and assume $U+V$ = $R$, and $U \cap V \subseteq J(R)$ .
Suppose that $v \in V$ and that $U + v$ is ...
0
votes
1answer
35 views
False proof of $R$ Noetherian, $I$ irreducible hence $I$ prime
Can you tell me what's wrong with my proof? Thanks.
Claim: If $R$ is a Noetherian ring and $I$ is an irreducible ideal in $R$ then $I$ is prime
Proof: Let $xy \in I$. We want to show that either ...
1
vote
1answer
65 views
Question about primary decomposition in Noetherian rings
I have a question about the following proof:
How do I get that $\mathfrak a$ is reducible? I thought perhaps one can argue that $\mathfrak a \cap \mathfrak a = \mathfrak a$ is a finite intersection ...
1
vote
2answers
67 views
Question about factor rings
Assume $m_i$ are maximal ideals in a ring $R$.
Then I have $m_1 \cdot \dots m_{k}$ is an ideal in $m_1 \cdot \dots m_{k-1}$ hence I can quotient to get a factor ring $m_1 \cdot \dots m_{k-1} / m_1 ...
1
vote
1answer
72 views
A question on the submodule given by action of an ideal
Suppose $M$ is a completely reducible left $R$-module for a ring $R$ and $I$ is an ideal of $R$.
Then prove that the following are equivalent:
$M=IM$
If $I$ annihilates an element $x\in M,\text{ ...
5
votes
1answer
150 views
A Gröbner Basis Computation Gone Bad
Here is the problem statement:
Consider the polynomial ideal $I = \langle b-r_1-r_2, c-r_1r_2 \rangle \subset \mathbb{Q}[r_1,r_2,b,c].$ Show that $I \cap \mathbb{Q}[b,c] = \langle 0 \rangle$.
...
6
votes
1answer
132 views
Question about proof of Going-down theorem
I have written a proof of the Going-down theorem that doesn't use some of the assumptions so it's false but I can't find the mistake. Can you tell where it's wrong?
*Going-down*$^\prime$: Let $R,S$ ...
4
votes
2answers
116 views
Insight of some concepts in commutative algebra
I really enjoyed the basic algebra course and wanted to teach myself a little more. So I am trying to learn commutative algebra from Atiyah-MacDonald and Eisenbud.
The department in our university ...
3
votes
1answer
179 views
Does every Noetherian ring contain at least one maximal ideal
I want to proof that a noetherian ring $R \neq \{0\}$ contains at least one maximal ideal.
My idea is to consinder $\langle 0 \rangle$ and $\langle 1 \rangle$:
If there is no ideal $I$ with $\langle ...
7
votes
3answers
633 views
A maximal ideal is always a prime ideal?
A maximal ideal is always a prime ideal, and the quotient ring is always a field. In general, not all prime ideals are maximal. 1
In $2\mathbb{Z}$, $4 \mathbb{Z} $ is a maximal ideal. ...
3
votes
0answers
121 views
Question about proof of Going-down theorem and prop. 3.16 in AM
Prop. 3.16 tells us that if $f: A \to B$ is a ring homomorphism and $\mathfrak p$ is a prime ideal of $A$ then $\mathfrak p$ is the contraction of a prime ideal of $B$ if and only if $\mathfrak p^{ec} ...
3
votes
3answers
311 views
Find an ideal in $K[x,y]$ that is maximal but not principal.
Let $K$ be a field. Find an ideal of $K[x,y]$ that is maximal but not principal. Prove your claims.(Here we are working in a commutative ring with $1\neq 0.$)
My idea:
Choose $K=\mathbb{Q}.$ ...
1
vote
0answers
97 views
Proof of going-up theorem
Can you tell me if my proof is correct? Thanks. (I'm using propositions 5.6 and 5.10 from Atiyah-Macdonald which I proved separately.)
Theorem: Let $R$ be integral over $S$. Let $p_1 \subset p_2$ be ...
3
votes
2answers
53 views
Proof of $R/I$ integral over $S/(S \cap I)$
Can you tell me if my reasoning is correct?
I want to prove if $S \subset R$ are rings and $R$ is integral over $S$ and $I$ is an ideal of $R$ then $R/I$ is integral over $S/ (S\cap I)$.
Let $R$ be ...
1
vote
3answers
71 views
proof for members of ideals
If $I$ is an ideal, could you show that if $ x\in I$ and $y\notin$ I, then $x+y \notin I$? It seems like an intuitively obvious statement and yet my rigor is failing me. So if you could show me all ...
3
votes
2answers
128 views
Counter-example that $I\cup J$ in a ring $R$ may not be an ideal
I've been doing some reading about ideals and here is another question (to which I couldn't yet find or construct a counterexample).
Let $I, J$ be ideals in a ring $R$. Then $I\cup J$ is contained ...
2
votes
5answers
355 views
The image of an ideal under a homomorphism may not be an ideal
This is an elementary question about ideals. Consider a ring homomorphism
$$
f: \mathbb{Z} \rightarrow \mathbb{Z}[x],
$$
and consider the ideal $\left< 2\right>$ in $\mathbb{Z}$. When why is ...
1
vote
1answer
83 views
Ideal generated by differential forms
I have troubles picturing what elements belong to a particular ensemble.
Let $\omega_1$,...,$\omega_r$ be differential 1-forms on a $C^\infty$ n-manifold that are independent at each point. ...
5
votes
0answers
137 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 ...
2
votes
1answer
177 views
Example of non-decomposable ideal
An ideal $I$ of a commutative unital ring $R$ is called decomposable if it has a primary decomposition.
Can you give me an example of an ideal that is not decomposable? All the examples I can think ...
1
vote
1answer
98 views
Explanation of passage in Atiyah-MacDonald
On page 52 they write "...By (4.3) we can achieve (i)..."
where (4.3) is the lemma on the previous page that states that if $q_i$ are all $p$-primary then $\bigcap_i q_i$ is $p$-primary and (i) is ...
0
votes
2answers
117 views
Do we have $(R/I)/(J/I) \cong (R/J)/(I/J)$?
Do we have $(R/I)/(J/I) \cong (R/J)/(I/J)$ where $R$ is a ring and $I,J$ are ideals?
If $I \subset J$ then it follows from the third isomorphism theorem.
In $R= \mathbb Z$ with $I = 3 \mathbb Z$ ...
1
vote
3answers
64 views
If $a$ in $R$ is prime, then $(a+P)$ is prime in $R/P$.
Let $R$ be a UFD and $P$ a prime ideal. Here we are defining a UFD with primes and not irreducibles.
Is the following true and what is the justification?
If $a$ in $R$ is prime, then $(a+P)$ is ...
2
votes
3answers
452 views
Ideal of the twisted cubic
The twisted cubic is the image of the morphism $\phi : \mathbb{P}^1 \to \mathbb{P}^3 , (x:y) \mapsto (x^3:x^2 y:x y^2:y^3)$, it is given by $X = V(ad-bc,b^2-ac,c^2-bd)$. Now I would like to compute ...
2
votes
1answer
141 views
Nilradical of a primary ideal is a minimal prime
I'd like to show the following claim:
The radical of a primary ideal $\mathfrak q$, $r(\mathfrak q)$, is the smallest prime ideal containing $\mathfrak q$.
Can you tell me if my proof is correct:
...
4
votes
1answer
69 views
Is there a nice way to classify the ideals of the ring of lower triangular matrices?
Suppose $T$ is the subset of $M_2(\mathbb{Z})$ of lower triangular matrices, those of form $\begin{pmatrix} a & 0 \\ b & c\end{pmatrix}$. So $T$ is a subring. Now I know that the ideals of ...
3
votes
1answer
120 views
Is there a distributive law for ideals?
I'm curious if there is some sort of distributive law for ideals. If $I,J,K$ are ideals in an arbitrary ring, does $I(J+K)=IJ+IK$? The $\subset$ containment is pretty clear I think. But the opposite ...
4
votes
1answer
69 views
Is there a ring with the lattice of ideals isomorphic to $(\omega+1)^{\operatorname{op}}?$
In this question, I gave an example of a ring whose lattice of two-sided ideals is order-isomorphic to $\omega+1$. I've been playing a bit with trying to find rings with a given lattice of ideals ...
2
votes
1answer
68 views
Diamonds of ideals, part 3
I'd like to wrap up the line of questioning started first in this question and then continued in this question.
The only variant left to try is:
"How close can you get to the Diamond lattice ...
2
votes
0answers
116 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 ...
1
vote
1answer
127 views
Every nonzero $x\in\mathbb{Z}[\sqrt{35}]$ belongs to finitely many ideals
How can one show that every nonzero element $x$ of the ring $\mathbb{Z}[\sqrt{35}]$ is contained in finitely many ideals? It is obvious in case of $x$ being invertible, but a general case is out of my ...
2
votes
1answer
129 views
Ring of analytic functions on the circle
Let $A = C^\omega(S^1)$ (resp. $C^\omega_{\mathbb C}(S^1)$) the ring of real-analytic real-valued (resp. complex valued) functions on the circle.
These rings have maximal ideals $\mathfrak m_p = ...
0
votes
2answers
170 views
Radicals of homogeneous ideals over semigroup-graded rings.
In this post I set out to prove an equation holding in a ring $R$ graded over $\mathbb{Z}$ or $\mathbb{N}$.
The equation was: $\sqrt{J^\ast}=(\sqrt{J})^\ast$, where $J$ is an ideal of $R$, $J^\ast$ ...
1
vote
1answer
291 views
Saturated ideal
Let $k$ be a field, let $I \triangleleft k[X_1,\dots,X_n]=S$ be an ideal and fix $f \in S$.
The saturated ideal of $I$ is $I^{sat}=I:f^\infty=\{g \in S \mid \exists m \in \mathbb{N} \ s.t. \ f^mg \in ...
3
votes
1answer
164 views
Is every proper nontrivial ideal in a Noetherian ring not flat?
I guess my general question is exactly what's in the title, but let me explain why I'm asking and how I came to it.
Consider the ideal $I=\langle x,y \rangle \subset k[x,y]$ for a field $k$. Just to ...
4
votes
3answers
76 views
For an ideal $I$ of $K[x]$, $K[x]/I$ is finitely generated iff $I$ is nonnull
In a book on rational series, a blunt statement is made to the effect that:
For $K$ a field, $I$ an ideal of $K[x]$, $K[x]/I$ is finitely generated iff $I$ is nonnull.
The statement elaborates ...
2
votes
2answers
249 views
The going-up theorem
I am reading Introduction to Commutative Algebra / Atiyah & Macdonald,
Theorem 5.11 ("Going-up theorem").
The statement is:
Let $A \subset B$ be rings, $B$ integral over $A$;
let $p_1 ...
