This tag is for questions about rings, which are a type of structure studied in abstract algebra and algebraic number theory.
21
votes
0answers
338 views
A short proof for $\dim(R[T])=\dim(R)+1$?
If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and nontrivial ...
9
votes
0answers
133 views
Maximal ideal space of $c_{\mathcal{U}}$
Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define
$$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$
which is a C*-algebra. Is there an accessible topological ...
9
votes
0answers
212 views
Is every model of modular arithmetic either even or odd?
Modular Arithmetic (MA) has the same axioms as first order Peano Axioms (PA) except $\forall x (Sx \ne 0)$ is replaced with $\exists x(Sx = 0)$.
...
9
votes
0answers
180 views
Minimal spectrum of a commutative ring
Can anyone explain to me why the minimal prime ideals of a commutative ring (with the subspace topology inherited from the Zariski topology) form a totally disconnected space, or give a reference? I ...
8
votes
0answers
108 views
Tensoring is thought as both restricting and extending?
I hope these questions are not too trivial.
Let $I$ be an ideal in $R$. Write $I'\subseteq R[t]$. Then the notion of tensoring
$$
(R[t]/I')\otimes_{\,\mathbb{C}[t]} \mathbb{C}[t]/\langle t-c ...
7
votes
0answers
105 views
Ideal in an Artinian Ring $I=aR=Rb$, prove $I=Ra=bR$
Let $R$ be an Artinian Ring and suppose there exists $a,b\in R$ s.t. $I=aR=Rb$, then prove $I=bR=Ra$.
(You may assume that a right Artinian Ring is Right Noetherian).
I've managed to get $Ra$,$bR$ ...
7
votes
0answers
152 views
Rings satisfying “for all $a \neq 0$, there is nonunit $b$ with $a+b$ a unit”
Consider the following condition on a ring:
For every nonzero $a$, there is a nonunit $b$ with the property that $a+b$ is a unit.
Observe that if $a$ is already a unit, then $b=0$ will do just ...
7
votes
0answers
116 views
Minimal systems of generators for commutative rings
Let $S$ be some base ring (a commutative ring or even just a field), and $R$ a commutative ring containing $S$ which is finitely generated (as an algebra) over $S$. What conditions guarantee that any ...
6
votes
0answers
70 views
An example of a compact multiplicatively unbounded ring
My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
6
votes
0answers
99 views
On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.
I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
6
votes
0answers
114 views
Is there an upper bound to the number of rings that can be obtained from a semigroup with zero by defining an additive operation?
Let $\mathscr S$ be the class of all semigroups with zero. For $(S,\times,0)\in\mathscr S,$ I want to count additive operations $+$ on $S$ such that $(S,+,\times,0)$ is a ring (possibly without ...
5
votes
0answers
57 views
+100
Amenable group rings embeddable in skew fields
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent:
(1) the group ring $K[G]$ is a domain;
(2) $K[G]$ is ...
5
votes
0answers
90 views
An example of a commutative ring in which every primary ideal is prime
It is clear that every prime ideal in a commutative ring is primary. The converse is false; for example, in the ring $\mathbb{Z}$ the ideal $(p^2)$ is an example of a primary ideal that is not prime ...
5
votes
0answers
82 views
Hilbert symbol over a ring
Normally the Hilbert symbol over a field $\mathbb{F}$ is defined for $a,b\in\mathbb{F}^*$ as follows:
$$ (a,b)=\begin{cases}1,&\text{ if }z^2=ax^2+by^2\text{ has a non-zero solution }(x,y,z)\in ...
5
votes
0answers
93 views
Non-reflexive module isomorphic to its double dual
Could you give me an example of a non-reflexive module isomorphic to its double dual?
I found an example here but I cannot understand it, do you have any simpler examples?
By this question we should ...
5
votes
0answers
217 views
System of polynomial equations over rational field
Fix $n\geq 2$. Let $p:=x_1^2+\ldots+x_{n-1}^2+1\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$. Suppose $u_1,\ldots,u_n,v_1,\ldots,v_n\in\mathbb{Q}[x_1,\ldots,x_{n-1}]$ satisfy the following equations:
...
5
votes
0answers
95 views
A characterization of $M_n(\mathbb{Z})$?
I am interested in rings $R$ with the following properties:
(1) $R$ is a free $\mathbb{Z}$-algebra of finite rank
(2) each two-sided ideal of $R$ is generated by an integer
The matrix rings ...
5
votes
0answers
113 views
Is there a simple example of a ring that satifies the DCC on two-sided ideals, but doesn't satisfy the ACC on two-sided ideals?
It follows from the Hopkins–Levitzki theorem that if a ring satisfies the DCC on left ideals, then it also satisfies the ACC on left ideals. I've been trying to find a counterexample to the following ...
5
votes
0answers
157 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
0answers
54 views
Artinian rings are perfect
Is there a simple way to prove that an Artinian ring is perfect? (in the commutative case)
4
votes
0answers
56 views
Deciding whether or not a class of modules is “big enough”
For the last few days I'm pondering the following question. The situation is this: $R$ is a commutative ring and $A$ a (noncommutative) $R$-algebra. I have a class $\mathcal{C}\subseteq\coprod_{S} ...
4
votes
0answers
52 views
A question on an answer on Math Overflow about Artin approximation
I have a question on an answer of this Math Overflow question.
Let $(A,I)$ be a commutative excellent normal local domain. The completion
$$
\hat A=\underset{\longleftarrow}{\operatorname{lim}} ...
4
votes
0answers
25 views
An explicit $\Lambda_R^\ell(M)$ when $M$ is not free
Let $\Lambda_R^n(M)$ be the nth exterior power of an $R$-module $M$. Let us assume $M$ is finitely generated.
When $M$ if free, say, $M=R^{\oplus d}$, we know \begin{equation}
\Lambda_R^n(M)\cong ...
4
votes
0answers
137 views
Defining multiplication on a Koszul complex
Let $R$ be a Nothearian commutative ring and $x$ and $y$ two elements in $R$. We construct the Koszul complex on $x$ and $y$. We start by the following two chain complexes:
$$
C_2=0\to ...
4
votes
0answers
98 views
Computing the ring of integers of a number field
This question arose from the need to see the splitting behaviour of primes over an extension of number fields. One criterion is the Kummer's theorem, which gives the decomposition of the base prime in ...
4
votes
0answers
78 views
Some elementary facts
What is the simplest and the most conceptual proof of some basic facts on algebraic geometry?
1) Hilbert's Nullstellensatz
2) Regular functions on projective variety - only constants
3) elemination ...
4
votes
0answers
54 views
If $A \hookleftarrow B \to R$ each contain $R$, is $R\to A\otimes_B R$ injective?
In this question, all rings and algebras are commutative with identity.
Let $R$ be a ring, and let $A$ be an $R$-algebra with an $R$-subalgebra $B$. Suppose that we have an $R$-algebra homomorphism ...
4
votes
0answers
169 views
Proof of Smith Normal Form as a Generalization of Rank-Nullity Theorem
For any matrix A with entries in a PID, there exist invertible matrices P and Q such that B = PAQ, where B is in Smith normal form. This theorem is usually proved by using elementary row/column ...
4
votes
0answers
88 views
An algebraic algorithm for finding inverses in the group algebra
This is an extension to my earlier question.
Is there a purely algebraic algorithm to find inverses in the group algebra? For example, in the group algebra $\mathbb{C}S_{4}$, how would one go about ...
4
votes
0answers
315 views
Relations between semi-artinian and $\pi$-regular rings
A ring $R$ [associative, with 1, not necessary commutative] is said to be right semi-artinian if every non-zero module over $R$ has a simple submodule.
A ring $R$ is said to be strongly $\pi$-regular ...
3
votes
0answers
53 views
Integral homomorphism induces a closed map on spectra
I'm trying to prove the following:
Let $f:A\rightarrow B$ is a integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow ...
3
votes
0answers
45 views
GCD in a subring is GCD in a bigger ring
Let $R$ be a UFD which is a subring of an integral domain $S$. If $r_1$ and $r_2$ are two nonzero elements of $R$ with GCD $d$, is it true that $d$ is also a GCD of $r_1$ and $r_2$ in $S$?
I know ...
3
votes
0answers
117 views
find a special element in group algebra
Let $G=\langle x, y, z| xyx^{-1}=zy, xzx^{-1}=z, yz=zy\rangle$, denote $l^1(G)^{\times}$ to be the set of units in $l^1(G)$, which we have considered as a ring with multiplication defined by the usual ...
3
votes
0answers
120 views
Polynomial rings in an arbitrary set of indeterminates
Recently I was reading an article where the author described a strange polynomial ring that I had never seen before. Here it is (I changed some words):
$T$ is a set (in the context of the article ...
3
votes
0answers
55 views
Hilbert’s zeros theorem, an application. (The algebraic variation)
Theorem: (Hilbert) If $k$ is a field, $A$ is a finitely generated $k$-algebra, and $M$ is a maximal ideal in $A$, then the factor $A/M$ is a finite extension of $k$. In particular if $k$ is ...
3
votes
0answers
35 views
Ring of holomorphic functoins
Let $\ O_n =\{\text{all holomorphic functions around the origin in} \Bbb C^n\}$, I'm trying to prove the follwoing, if $f_i=z^2-w^{n_i},i=1,2$. then$$\frac{O_2}{f_1}\simeq\frac{ O_2}{f_2}$$if and only ...
3
votes
0answers
58 views
When is $\mathbb{Z}\Gamma$ a left Noetherian ring?
Denote $\Gamma$ to be a countable discrete group, let $\mathbb{Z}\Gamma$ to be its integer group ring.
A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals.(c.f. ...
3
votes
0answers
74 views
What breaks down in linear algebra over the rings (or commutative rings)?
Let $R$ be a ring (with $1$). Last night, I was trying to prove that $M_{n}(R)$ (the ring of $n \times n$ matrices over $R$) is a ring. As I have done in my previous linear algebra course (which was ...
3
votes
0answers
119 views
In a PID without unit an ideal is maximal iff it is prime.
As the titles says, I need to show that in a PID $R$ an ideal is maximal iff it is prime. This is easy to do if $R$ has a multiplicative identity. I can not do it if $R$ does not have an identity. It ...
3
votes
0answers
107 views
When is intersection of infinitely many maximal ideals zero?
I've been trying without success to figure out what are the rings $R$ such that whenever $M_n, n \in \omega$ is a countably infinite collection of pairwise distinct maximal ideals then $\bigcap_{n \in ...
3
votes
0answers
78 views
On complexes of projective modules
How can I prove the following statement?
Let $\beta: B\rightarrow C$ be a quasi-isomorphism of complexes of $R$-modules. If $P$ is a complex of projective $R$-modules which is bounded below, then ...
3
votes
0answers
61 views
Applications of Govorov-Lazard Theorem?
The Govorov-Lazard Theorem states that a (right) module over an unital ring is flat iff it is a direct limit of finitely generated free (right) modules.
I wonder if this theorem has interesting ...
3
votes
0answers
102 views
Extension of the theorem of Jacobson
Let $A$ be a ring. Let $E$ be the set of polynomials $\{X^n-X \in \mathbb{Z}[X]|n \in \mathbb{N}^*-\{1\}\}$.
By the theorem of Jacobson, we know that if for each $a\in A$ there is an element of $E$ ...
3
votes
0answers
82 views
3
votes
0answers
142 views
Coproduct in the category of (noncommutative) associative algebras
For the case of commutative algebras, I know that the coproduct is given by the tensor product, but how is the situation in the general case? (for associative, but not necessarily commutative algebras ...
3
votes
0answers
45 views
Does the singleton reduction system $\{(x^2y,yx)\}$ lead to a normal form?
Suppose you have a singleton reduction system $\{(x^2y,yx)\}$. Does such a system lead to a normal form on the corresponding $k$-algebra $k\langle x,y\rangle$, where $k$ is a commutative, ...
3
votes
0answers
174 views
Noetherian rings, why commutativity?
I am looking for an answer to why one has to assume commutativity of a ring $R$ in proving some results about Noetherian rings. For example, Let $R$ be a commutative ring; look at the proof(s) of the ...
3
votes
0answers
141 views
Projective modules over rings without unit
For rings with unit there are at least three ways to define a projective module:
The universal property, i.e. $P$ is projective if for any epimorphism $M\to N$ and any morphism $P\to N$ there exists ...
3
votes
0answers
273 views
Ring with subring isomorphic to $\mathbb{Z}$ and subring isomorphic to $\mathbb{Z}_{3}$
This is a homework question that I'm either not thinking through all the way, or I'm overcomplicating the issue. It reads
Give an example of a ring that contains a subring isomorphic to ...
3
votes
0answers
179 views
Show $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to $\mathrm{E}(\mathbb{Z}_{m})\times\mathrm{E}(\mathbb{Z}_{n})$ if and only if $(m,n)=1$
Define $\mathrm{E}(\mathbb{Z}_{i})$ to be the group of invertible elements of the ring with unity $\mathbb{Z}_{i}$.
Show that $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to ...
