Questions tagged [ring-theory]
This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.
22,262 questions
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A module is artinian and noetherian iff it has finite length
As stated in the title, I am trying to prove that (I am assuming $R$ is a commutative ring with unity);
Let $M$ be an $R$-module. Then $M$ is artinian and noetherian if and only if $\ell \left( M\...
- 195
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On hypersurface singularity.
Suppose that $A \colon= {\Bbb C}[X,Y,Z,W]/f(X,Y,Z,W)$ is a three-dimensional normal ring over ${\Bbb C}$. I am seeking for $f(X,Y,Z,W)$ such that ${\mathrm{Spec}}\,A$ has singularities along a curve $...
- 1,017
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Which integral domains $R$ are flat over every subring?
It is well-known that if every integral domain containing a given integral domain $R$ is flat over $R$, then $R$ is a Prüfer domain. So I would like to ask the question about the other direction: ...
- 2,781
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The "$aRa$" Definition of Semiprime Ideals Implies the "One-Sided Ideal" Definition
For a noncommutative ring $R$ (and let's also say not necessarily with 1), there are 4 equivalent ways to say that an ideal $K$ is semiprime:
If $A$ is a 2-sided ideal of R with $A^n \subset K$, then ...
- 105
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Prove that constant times polynomial is zero, given product of two polynomial is zero. [duplicate]
Problem statement: Suppose $R$ is a commutative ring, and $f$ is a nonzero polynomial in $R[x]$. Suppose there exists $g$, another nonzero polynomial in $R[x]$, and $f\cdot g=0$, prove that there ...
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T is an order implies T is a maximal preorder
Let $A$ be a ring, a subset $T\subset A$ is called a preorder if
$-1\not\in T$
$T+T\subset T$
$T\cdot T\subset T$
$A^2:=\{a^2\mid a\in A\}\subset T$
$T$ is an order of $A$ if additionally
$A=T\cup -...
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Example of a ring having a number $m$ of prime ideals and a number $n$ of maximal ideals
How to construct an example of a ring having a number $m$ of prime ideals and a number $n$ of maximal ideals ?
Please instruct me how to think this example step by step. Thanking you beforehand!!!
- 334
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2
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Proving every two composition series of a module are of the same length (half of the *Jordan-Holder theorem*)
This is a second followup question to this question I asked a couple of days ago (here is the first followup question). After resolving the issues I raised in both of the linked questions I proceeded ...
- 195
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localization of ring and module
Let $R$ be a commutative ring with $1$, and let $S$ be the set of elements that are not zero divisors in $R$; then $S$ is a multiplicatively closed set. Hence we may localize the ring $R$ at the set $...
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Finite division rings that are non-associative yet commutative [closed]
Do finite non-associative fields exist? By “non-associative field”, I mean a set $S$ equipped with two binary operations $+$ and $\cdot$ such that
$S$ with $+$ is an Abelian group
$S/\{0\}$ with $\...
- 167
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many example of rings [closed]
I'm studying commutative ring theory from Hideyuki Matsumura's book, but it's so abstract that I try to come up with lots of concrete examples on my own---for instance, classifying commutative ...
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Problems with the proof of $\mathbb{Z}[x]/\langle 3, x^{2} + 1 \rangle$ is a field [duplicate]
I see some similar problems, but I'm stuck on what I want to prove.
For prove that $\mathbb{Z}/\langle 3, x^{2} + 1 \rangle$ is a field is enough to prove that $\langle 3, x^{2} + 1 \rangle$ is ...
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If $A \to B$ is universal with respect to sending $I$ to the unit ideal and $A$ is an integral domain, does $B$ also have to be an integral domain?
Let $R$ be a commutative ring with unity. Let $A$ be a commutative $R$-algebra. Let $I \subseteq A$ be an ideal with $I \neq \{0\}$. Let $\varphi: A \to B$ be a map of $R$-algebras satisfying the ...
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Literature on $\mathrm{Ext}_R(-,-)$ as simultaneous $\delta$-functor in each variable
This is a followup question to Balancing Ext as a δ-functor. In the original post, I realized that $\mathrm{Ext}^n_R$ is an additive bifunctor with $\delta$-functions for exact sequences in each ...
- 7,713
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The multiplication of fractions of non-commutative ring is well-defined
In Lam's Lectures on modules and rings, page 302 (proof of Theorem 10.6), he defines the fraction of a ring $R$ with a multiplicative set $S$ which satisfies the following conditions:
Right Ore ...
- 398
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Confusion in Localization Section of Bosch's Commmutative Algebra
I'm rereading the section of Bosch's commutative algebra book concerning localization, and I am a bit concerned about the definitions he uses. He defines $R_S$ as the set of all fractions of the form $...
- 335
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Adjointness of Hom and Tensor: understanding statement
A statement made in the class was as follows:
Theorem Let $A,B$ be rings (may be non-commutative). Let $X_B$ be right $B$-module. Let ${}_BY_A$ be $(B,A)$-bimodule. Let $Z_A$ be right $A$-module. ...
- 3,485
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$M$ is a $\mathbb{F} [x]$ module iff $M$ is a $\mathbb{F}$ vector space with a linear map $\varphi : M \to M$ s.t. $f\ast v = [f(\varphi)](v)$
I aim to prove the following claim:
Claim - Let $\mathbb{F}$ be a field and let $R = \mathbb{F} \left[x \right]$ be the polynomial ring over $\mathbb{F}$. Then $M$ is a $R$-module if and only if $M= ...
- 889
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Is there a semi-local PID, other than fields and DVRs?
Is there a semi-local PID, other than fields and DVRs?
I know local PIDs must be fields or DVRs.
If semi-local PIDs are also fields or DVRs, I would like to see a proof.
If not, could you give a ...
- 3
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What is the $\mathbb{Z}$-grading on $\mathbb{C}[x_{ij},\mathrm{det}^{-1}]$?
Perhaps this is a basic questions, but I would like to understand the $\mathbb{Z}$-grading on $\mathbb{C}[x_{ij},\mathrm{det}^{-1}]$. I understand that the ring
$
\mathbb{C}[x_{ij},t]
$
is graded with ...
- 743
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How to show symbolic adjunction of roots does not depend on the order
Consider $p(t) = (t^2 -2)(t^2 -5)$ in $\mathbb{Q}[t]$. Depending on the order you adjoin formal square roots of $2$ and $5$ to $\mathbb{Q}$, you get two different splitting fields,
$$(\mathbb{Q}[t]/(t^...
- 1,177
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Subring generated by $2$ in $\mathbb{Z}$: $2\mathbb{Z}$ or $\mathbb{Z}$?
I'm confused about the subring generated by an element in $\mathbb{Z}$.
Consider the element $2 \in \mathbb{Z}$. I defined the subring generated by $2$ as:
$$\langle 2 \rangle = \left\{ \sum_{i=0}^{n} ...
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3
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What is the complete list of 3-dimensional real associative unital algebras?
Are there any besides $\mathbb{R}^3, \mathbb{R}\times \mathbb{C}, \mathbb{R}[\epsilon]/\epsilon^3, \mathbb{R}\times\mathbb{R[\epsilon]}/\epsilon^2$? Are there any non-commutative?
Regarding the 4-...
- 10.6k
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Uniqueness of localization of an integral domain in a common field
Let $R$ be an integral domain, $S \subset R$ a multiplicative system.
We assume that there exists a field $K$ such that $R$ is a subring of $K$.
Then
$$ B = \{ as^{-1} \ \mid \ a \in R, s \in S\} $$
...
- 57
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Structure of the quotient ring $R/\mathfrak{m}^r$ [closed]
Let $R$ be a commutative ring, $\mathfrak{m}$ be one of its maximal ideals. It is well known that $R/\mathfrak{m}$ is a field. I wonder if this result could be generalized (properly weakened, of ...
- 1,229