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.

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14 views

Flat, non-faithfully flat, submodule of a faithfully flat module

Let $R$ be a unital ring (not necessarily commutative) and let $M$ be a (left) faithfully flat module over $R$. Can there exist a non-zero flat (left) $R$-submodule of $M$ that is not faithfully flat.
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18 views

Does $\left(\prod_iM_i\right)/\left(\prod_iM_ir\right)\cong \prod_{i\in I}\left(M_i/M_ir\right)$ hold for modules?

Let $R$ be a ring, $r\in R, \{M_i\}_{i\in I}$ a family of $R$-modules and $\{M_ir\subseteq M_i\}_{i\in I}$ be a family of $R$-submodules. Does the following hold true $$\left(\prod_iM_i\right)/\left(...
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12 views

Cayley graph over ring structure

I am currently working on algebraic graph theory and would like to know whether Cayley graph over ring structure has been defined in any of the research papers. I tried finding but I could not get any ...
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1answer
31 views

If $M$ is a Noetherian $R$-module, every non-empty set of proper submodules of $M$ has a maximal element

My proof: Suppose for a contradiction that we have set of proper submodules of $M$, let's call it $\mathfrak{M}$, wihch is non-empty and does not have a maximal element. Choose $N_1 \in \mathfrak{M}$, ...
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11 views

Ideal on polynomial ring generated by prime ideal on the original ring

I want to prove that if $R$ is a communtative ring with unity and $I$ a prime ideal of $R$, then $$J=(I,X)$$ is a prime ideal on $R[X]$ I know that the ideal generated by $I$ in $R[X]$ is prime but I ...
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1answer
18 views

Does taking covariant Hom commute with taking (co-)kernel?

Let $A$ be a Commutative ring and $R$ be a commutative $A$-algebra. So every $R$-module has a natural $A$-module structure, and for every $A$-module $W$, and $R$-module $M$, the $A$-module $\text{Hom}...
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1answer
18 views

Max number of ideals in a ring

Find the maximum number of ideals in a ring $R$ with $4$ elements. So far I managed to see that if our ring consists of $(0,b,c,d)$ then if we want to have that $(0,b),(0,c),(0,d)$ are ideals then if ...
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0answers
41 views

Maximal Ideal problema

Let $M_{2,2}(\Bbb Z)$ be the ring of $2 \times 2$ matrices with integer entries and $S=\left\{ \begin{pmatrix} a & 2b \\ 2b & a \end{pmatrix} \mid a,b \in \Bbb Z \right\}$ its subset. I need ...
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1answer
76 views

The two rings are not isomorphic - why?

I’m proving the following rings are not isomorphic $$ \mathbb{K}[x,y]\not\cong\mathbb{K}[x,y,z]/(xy-z^2) $$ $\mathbb{K}$ is just an algebraic closed field I can see they are not isomorphic since there ...
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18 views

Does every evaluation map $g: \Bbb{Z}[X,Y]/(\ker \pi) \to \Bbb{Z}$ factor some evaluation map $f:\Bbb{Z}[X,Y] \to \Bbb{Z}$?

Let $\Bbb{Z}[X,Y]$ be the polynomial ring. I know that every evaluation ring hom $f: \Bbb{Z}[X,Y] \to \Bbb{Z}$ is determined by where you send $X$ and $Y$. Let $I = (X^2 - Y^3)$ for example, but it ...
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29 views

Zero divisors in the Dirichlet ring

I'm trying to determine if the ring $(\mathbb{A}, +, *)$ is an integral domain, where $\mathbb{A}$ is the set of arithmetic functions and $*$ is the Dirichlet convolution. To do so, I'm trying to ...
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2answers
25 views

$\overline{x}$ is nilpotent in $R[x]/\langle x^n-a\rangle$ where $a$ is nilpotent in $R$

[Dummit and Foote, Exercise 7.4, problem 14.(d)] If $f(x) = x^n - a$ for some nilpotent element $a \in R$, prove that $\overline{x}$ is nilpotent in $R[x]/(f(x))$. To show that $\overline{x}$ is ...
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1answer
33 views

Given a ring with four elements, find the maximal number of elements in an ideal

Given a ring with four elements. What is the maximum number of elements that an ideal of this ring can have? I think that the answer is $4$ since if $R$-the ring has an identity element, say $e$ and ...
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0answers
29 views

When does a monomorphism must be an automorphism?

Say R is a ring, and denote $\varphi :R\rightarrow R$ a one-to-one homomorphism. Consider the case where $\varphi$ is not onto. One example is the mapping $(a_{0}, a_{1}, ...) \mapsto (0, a_{0}, a_{1},...
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30 views

Completion of DVR is again DVR?

Let $R$ be a DVR, that is, $1$-dimmensional Noether regular local ring. And $M$ be it's maximal ideal. My question: Is completion of $R$ with respect to $M$ is still DVR? I know completion holds krill-...
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2answers
60 views

Is the ideal $\langle x^2 + 1\rangle$ maximal in $\mathbb{Z}_3[x]$?

Is the ideal $\langle x^2 + 1\rangle$ maximal in $\mathbb{Z}_3[x]$? I am going about this by trying to prove that $\frac{\mathbb{Z}_3[x]}{\langle x^2 + 1 \rangle}$ is a field. I can prove commutative ...
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0answers
20 views

How can I prove that an invented operation works with negative numbers? [duplicate]

Just out of curiosity I tried making a new operation to see its properties. It is like a variation of addition that works like this: a ¨ b = a + b - 1 For example, I found that it has the commutative ...
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2answers
41 views

Showing $\langle x,y \rangle$ is a prime ideal of $K[x,y]$

The biggest thing that it stopping me from progressing in this question is the fact that I have two elements in the ideal, this topic it still very new to me. I can show that $I=\langle x \rangle$ and ...
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1answer
41 views

When are we allowed to cancel $a$ in an equation in a ring?

Given three elements $a,b,c$ in a ring $R$. When are we allowed to cancel $a$ from the equation $ab=ac$? My answer was when there is an element $a'$ which is the element $a$ raised to power $(-1)$, ...
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1answer
25 views

complete DVR which has the same residue field

Let $R$ and $R'$ be a DVR, both are complete ( with respect to its maximal ideal $M$), and $R$ and $R'$ are both have the same uniformizer $π$. We assume $R$ and $R'$ has the same residue (as a set). ...
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1answer
37 views

If $\operatorname{height}(P)=\operatorname{height}(J)$, then $P$ is minimal over $J$

Let $R$ be a regular local ring, and let $P$ be a prime ideal in $R$. Assume that $\operatorname{height}(P)=h$ and $P=(x_1,\ldots, x_k)$, for some $x_1,\ldots,x_k\in R$, and $k\geq h$. Let $J=(x_1,\...
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1answer
123 views

How do I show that if the product of idempotents is idempotent, then idempotents commute with other elements?

I was not able to prove the following theorem. I would appreciate some help regarding it: If the product of each two idempotent elements in a unity ring is an idempotent element itself, prove that ...
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1answer
28 views

$v(n!)=\sum_{i=1}^{\infty} [\dfrac{n}{p^i}]v(p)$

Let $R$ be a discrete valuation ring that is complete with respect to its maximal ideal $M$,and let $v$ be the valuation on $R$. And $p$ be a prime. And we assume $0<v(p)<∞$. Why equality $v(n!)=\...
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0answers
23 views

What is the educated way to find all maximal ideals of the ring?

Suppose I am given the ring, with basis elements $(e_1, e_2, \alpha, \beta)$. $e_1$ and $e_2$ are idemponents $e_1^2 = e_1, e_2^2= e_2$ and the following holds: $$ \alpha e_1 = \alpha = e_2 \alpha \...
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1answer
178 views

$u^{2}-1$ is a unit in a ring $A$

Hello I have the next question that I want to prove (or one counterexample if there is one) Let $A$ be a ring with maximal ideal $M$ and quotient field $k=A/M$ of size at least $4$ such that the ...
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1answer
28 views

Maximal ideals in certain affine algebras

Let $a_1,\ldots,a_m \in \mathbb{C}[x_1,\ldots,x_n]$, with $m > n$, and let $R=\mathbb{C}[a_1,\ldots,a_m]$. For example, $R=\mathbb{C}[x^2,x^3] \subseteq \mathbb{C}[x]$. Question. Could we find all ...
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1answer
25 views

Example of a ring of infinite uniform dimension

I'm trying to find an example of a ring of infinite uniform dimension. We say that a ring has uniform dimension $n$ if there exists $n$ non-zero ideals $I_{i}$ of $R$ such that $$ \bigoplus_{i=1}^{n} ...
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1answer
75 views

Ring, in which $1+1=0$ holds

First question comes from famous university's first examination of linear algebra. First question: Let $R$ be a finite ring, in which $1+1=0$ holds. Then, I would like to prove the number of elements ...
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1answer
31 views

Please prove that if a ring has a finite number of elements, then the characteristic of $R$ is a positive integer.

My proof is: Let $R$ have $n$ elements Let $x \in R$ If $G$ is a finite group and $H$ is a subgroup of $G$ then order of $ G = (\text{order of $H$})(\text{index of $H$ in $G$})$. Therefore, we get $o(...
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2answers
32 views

Is a localization at a maximal ideal of a polynomial ring a perfect ring?

There are several equivalent definitions for a perfect ring $R$ (not necessarily a commutative ring), for example: Every flat left $R$-module is projective; see wikipedia. Also, there is the notion of ...
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2answers
46 views

$M^n/M^{n+1}$ is vector space over a residue field $k=R/M$

Let $(R,M)$ be a local ring. I heard that $M^n/M^{n+1}$ is vector space over a residue field $k=R/M$. I would like to confirm this is true. Firstly, Could you tell me how we define map $k×M^n/M^{n+1}→...
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1answer
27 views

Sum and multiplication on graded algebras

Let $A$ be an algebra over a field $\mathbb{K}$ and $G$ a commutative group under addition. This algebra is called a $G$-graded if there exists a family $\{A_{g}\}_{g\in G}$ of vector subspaces of $\...
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1answer
26 views

What does the following interpretation of $k[x, y]/(y^2-x^3+x)$ mean?

$k[x, y]/(y^2-x^3+x)$ is described as being "something like polynomial functions on the curve $y^2-x^3+x$." I don't understand what this is supposed to mean. It is explained further that an ...
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2answers
43 views

Image of a maximal ideal $I$ by a surjective homomorphism $f$ such that $\ker f \subseteq I$ is maximal

Let $R$ and $S$ be rings, $I$ a maximal ideal of $R$, and let $f: R\rightarrow S$ be a surjective ring homomorphism such that $\ker f \subseteq I$. I’d like to show that the image $f(I)$ will be a ...
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6answers
75 views

What is a simple example for an exact sequence? Because in my opinion it seems that all the modules in the sequence should be equal to zero

Let $M_i$ be R-modules and $f_i$ be homomorphisms of R-modules If $\forall _n\ker f_n=\operatorname{im}f_{n-1}$, wouldn't that mean that for $$...\:\rightarrow M_{n-1}\:\rightarrow ^{f_{n-1}}\:M_n\:\...
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1answer
61 views

What exactly is a cokernel? What's the motivation behind that, and what's its use? And why is it a quotient module?

Having a homomorphism of $R$-modules $f\colon M \rightarrow N$, we say a cokernel is $N/(\operatorname{im} f)$ which is a $R$-module (quotient). Because $\operatorname{im} f\subset N$, this would mean ...
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1answer
27 views

Let $R$ be an arbitrary ring, $I\subset R$ an ideal. Why is $R/I$ an R-module?

We've defined that for any arbitrary ring $R$ and ideal $I\subset R$, $R/I$ is a module and is called a quotient R-module. But why is $R/I$ an R-module in the first place? I can't find anything on the ...
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1answer
22 views

Showing $x \in R$ is invertible if and only if $x$ is not contained in any maximal Ideal [duplicate]

Here $R$ is commutative. $\Rightarrow$ is fairly easy to prove as every Ideal containing a unit must be the entire Ring so it cannot be maximal. But I'm having some trouble with $\Leftarrow$, this is ...
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2answers
29 views

Prime Ideals in finite direct product of commutative Rings [duplicate]

Let $A_1, \dots , A_n$ be commutative unitary Rings and $A = \prod_{i=1}^{n} A_i$. Then every prime Ideal $\frak{p}$ $\subset A$ is of the form $\pi_i^{-1}(\frak{p}_i)$ where $\pi_i: A \to A_i$ are ...
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1answer
69 views

Copies of $\mathbb{C}$ in Artin-Wedderburn decomposition

Let $G$ be a finite group, and let $R = \mathbb{C}G$ be the group algebra. Show that the number of distinct group homomorphisms from $G$ to $\mathbb{C^*}$ equals the number of copies of $\mathbb{C}$ ...
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1answer
50 views

Define ring and field

An algebraic ring and field are defined by two inner binary operations. I am only unsure about one thing: do the operators have to be plus and times or are they arbitrary? Almost every book (and ...
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2answers
40 views

How to show this ring is Noetherian?

I got the following exercise: Let $W$ be a finite-dimensional $\Bbb{R}$-vector space. Let $\Bbb{R}_W=\Bbb{R}\times W$. Define addition and multiplication by $(r,w)+(s,v)=(r+s,w+v)$, $(r,w)*(s,v)=(rs,...
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2answers
52 views

How to show $\frac{R}{J} \subset \frac{R}{I}$ when $I \subset J$?

I know the elements of $\frac{R}{J}$ are like $r+J$ when $r \in R$. So I have to show that there is some $s \in R$ that $r+J = s+I$. But I don't know how to do that. I actually want to show that $\...
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0answers
25 views

Are prime ideals mapped to prime ideals?

Let $A$ and $B$ be two commutative rings and $\phi$ be homomorphism between them , if $P$ is a prime ideal of $A$, then is $\phi(P)$ a prime ideal of $B$? I think the statement is false, but ...
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0answers
24 views

An example of an ideal contained in union of ideals but not contained in any ideal seperatly [duplicate]

Find an example for a ring $R$ and ideals $I_1,...,I_n,J$ s.t $J\subseteq\bigcup_{i=1}^{n}I_i$ and $J\not\subseteq I_i$ for every $1\leq i\leq n$. I tried looking at some familiar rings like $\Bbb{Z},...
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1answer
33 views

Zariski topology closure of image of a homomorphism

Just started dealing with Zariski's topology on specta, and encountered the following question: $R,S$ commutative rings with unit and $\phi:R\rightarrow S$ homomorphism. Prove that the induced map $\...
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1answer
79 views

Let $R$ be a complete local ring with respect to maximal ideal $I$. Then, intersection of $I^n$ is zero.

Let $R$ be a complete local ring with respect to maximal ideal I. I would like to prove intersection of $I^n$, $n\ge1$ is zero. My attempt : $R$ is complete with respect to $I$, so $R$ is Hausdorff ...
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0answers
38 views

An example of graded rings

Let $R$ be a commutative ring, and $I$ an ideal in $R$. Then, we have a graded ring $R_{\ast}:= \bigoplus_{n\geq 0} I^n$, where $I^0=R$. So, let $A$ be a polynomial ring with a valuable $x$ over $\...
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0answers
43 views

Determining whether $ \langle 2 - 5\omega_3\rangle$ is a prime ideal of $\mathbb{Z}[\omega_3]$ [closed]

$R = \mathbb{Z}[\omega_3]$ is defined to be $\{a + b\omega_3 : a,b \in \mathbb{Z}\}$ where $\omega_3$ is the cube root of unity. I want to see if $I = \langle 2 - 5\omega_3\rangle $ is a prime ideal. ...
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1answer
64 views

Best notation for $\mathbb{Z}_m^n$ [closed]

This is an easy question. Which notation do you think is better for the set: $A:=\left\{x=[x_1,\dots,x_n]: x_i\in \mathbb{Z}_m\right\}$, where $\mathbb{Z}_m=\left\{i:0\leq i < m \right\}$. I've ...

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