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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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2answers
17 views

Idempotent elements in a modulo n ring

I'm trying to find the idempotent elements of the ring ($\Bbb Z_{36} $, +, $ \cdot $) so I "split" it into $ \operatorname{Idemp}(\Bbb Z_4 \times \Bbb Z_9) $, meaning $\operatorname{Idemp}(\Bbb Z_4) \...
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0answers
20 views

Find an ideal $I$ in $A$ so that $A/I$ is a finite field of $25$ elements.

Let $A = \frac {\Bbb Z[X]} {\left ( X^4+X^2+1 \right )}.$ Find an ideal $I$ in $A$ such that $A/I$ is a finite field of $25$ elements. I have seen that the polynomial $X^4+X^2+1$ is reducible in $\...
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1answer
28 views

Inverse of a finitely generated ideal in UFD

Let $R$ be a UFD and $K$ be its field of fractions. Let $A$ be an ideal of $R$. Define for this $A$, the $R$-submodule $A^{-1}$ of $K$ given by $$A^{-1}=\{ \alpha\in K \,\,:\,\, \alpha A\subseteq R\}....
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0answers
32 views

Yet another question about finding irreducible components : $(XY-Z^3,XZ-Y^3)$

Let $R=k[X,Y,Z]/J$ where $J=(XY-Z^3,XZ-Y^3)$ , $k$ a field not necessarily algebraically closed. I'm interested in computing: the minimal primes; I sketch here a tentative solution: A prime ...
0
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2answers
30 views

Characterization of pretty rings

Today I answered this question Characteristics of a pretty ring and wondered whether one could characterize these rings. Definition: A pretty ring $R$ is a ring with unity 1, not a field, and each ...
1
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1answer
14 views

Prove that $u-i$ is a maximal ideal of $\frac {\Bbb C[X,Y]} {\left (X^2+Y^2-1 \right )}.$

Let $A = \frac {\Bbb C[X,Y]} {\left (X^2+Y^2-1 \right )}.$ Let $u=X+iY.$ Show that $(u-i)$ is a maximal ideal of $A.$ Any ideal of $A$ is an ideal of $\Bbb C[X,Y]$ containing the ideal $\left ( X^2+...
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1answer
22 views

Can intersection of two different maximal ideals of Euclidean ring contain prime element?

Can intersection of two different maximal ideals of Euclidean ring contain prime element? We define $I$ maximal ideal of ring $R$, if there is no such ideal $I’ \neq R$, that $I \subset I’ \subset R$....
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1answer
25 views

Integrally closed domain preserved under prime quotient. Where is my mistake?

It is not true that for an integrally closed domain $A$ and any prime ideal $\mathfrak p$, the quotient $A/\mathfrak p$ is integrally closed as well. For example $\mathbb Z[x]$ is integrally closed, ...
4
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2answers
104 views

Characteristics of a pretty ring

This is a problem from a test I took today. Definition: A pretty ring $R$ is a ring with unity 1, not a field, and each nonzero element can be written uniquely as a sum of a unit and a nonunit ...
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4answers
74 views

Prove that 1 = 0 at the trivial ring

I need to prove that zero is one at the trivial ring, but I don't have yet that one is a member of the trivial ring (the only constant at my zero ring is zero). So I thought to prove first that, if R ...
0
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1answer
56 views

Dummit and Foote $(3^{ed})$ 7.4.13

Let $R$ be a ring with $1\neq 0$. Let $\varphi: R \rightarrow S$ be a homomorphism of Commutative Rings. If $P$ is a prime ideal of $S$, then prove that $\varphi^{-1}(P)=R$ or $\varphi^{-1}(P)$ is a ...
5
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1answer
59 views

How to write an arbitrary polynomial in $n$ variables

Be $k$ a field. I'm trying to define a function on $k[x_1, ..., x_n]$. However, I know of no way to write an arbitrary element of this ring efficiently. I read somewhere about using the $S_n$-orbit, ...
1
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1answer
26 views

Localization in a ring which is not an integral domain

In an integral domain $A$, localization by a prime ideal $\mathfrak p$ (obtaining the local ring $A_\mathfrak p$) can intuitively be thought of as simply formally inverting all the elements of $A\...
3
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3answers
44 views

Definition of $\mathbb{C}$-algebra and $\mathbb{C}$-algebra maps in Invitation to Algebraic Geometry

I am working through Karen Smith's An Invitation to Algebraic Geometry, and I am confused with the following problem from the text. Let $R$ be a $\mathbb{C}$-algebra, and let $I$ be an ideal of $R$. ...
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0answers
6 views

Trying to prove the symmetry of a relation in the construction of monoid of left fractions.

I am studying "Noncommutative localization in Noncommutative Geometry" by Zoran Skoda, and, given a monoid $R$ with a set of left denominators $S$, he is constructing a monoid of left fractions. ...
1
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1answer
16 views

show that a kernel is principal, here the application : $ \text{ev}_{x = t^2, y = t^3} $

$\mathbb C [x,y] \to \mathbb C [t]$ with the evaluation : $$ \text{ev}_{x = t^2, y = t^3} $$ How can you show that the kernel is principal in $\mathbb C[x,y] $? I think the kernel is the polynomial ...
0
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1answer
57 views

Converse of : If $M$ is Noetherian then $M_m$ is Noetherian for every $m\in\operatorname{Max}(R)$.

Let $R$ be a Noetherian ring and $M$ be an $R$-module. We know that if $M$ is Noetherian then $M_m$ is a Noetherian $R_m$-module for every $m\in\operatorname{Max}(R) $. Now, I want to know if the ...
1
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2answers
44 views

General intuition behind Ring isomorphism when it involves Polynomial Rings and help for proving it.

I want to ask in general but I will give an example that I am trying to prove. My question is how you can show one quotient or non quotient ring of polynomials is isomorphic to another ring/field. For ...
3
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1answer
30 views

Extension of $k$-vector space structure to $k[x]$-module structure

Be $k$ a field, $k[x]$ the polynomial ring over $k$, $V$ a $k$-vector space, and $f \in \text{End}_k(V)$. We want to see how the $k$-vector space structure can be extended to a $k[x]$-module ...
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0answers
19 views

Restrict the algebraic structure of the set of points without changing the points of the single coordinates

Take a point $p$ as an ordered pair of rational numbers ($\frac{x}{y}$, $\frac{x}{y}$). Consider that the set of all points $p$ of the Cartesian plane forms a field ℚ. What happens in the points $p$ ...
1
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1answer
29 views

Maximal Ideal Criteria in a Non-Commutative case

Question: Let $R$ be a non-commutative ring with $1\neq 0$. Let $M$ be an ideal(two-sided) of $R$. If $\frac{R}{M}$ is a field. Show that $M$ is a maximal ideal. My Attempt: Suppose that $M$ contains ...
2
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2answers
25 views

Prove that there exist $x_j, y_j, z_j ∈ A$ such that $a^{50}x_j +b^{20}y_j +c^{15}z_j = 1$.

Let $A$ be a commutative ring with $1$, and let $a, b, c \in A$. Suppose there exist $x, y, z ∈ A$ such that $ax+by +cz = 1$. Then there exist $x_j, y_j, z_j ∈ A$ such that $a^{50}x_j +b^{20}y_j +c^{...
1
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0answers
33 views

Automorphisms of Infinite Matrices

I was recently reading the following paper: Structure of Leavitt Path Algebras of Polynomial Growth by Alahmedi et al. and I have a question about one of the results. Fix a basis $\mathcal{B} = \{...
2
votes
2answers
47 views

What do the finitely generated free $R$-modules look like?

Let $R$ be a ring, maybe commutative or unital - but I do not know which condition is required. Then what are the finitely generated free $R$ modules? Does this imply $R \simeq \bigoplus_{i \in I} ...
2
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0answers
32 views

Show that, for $n \in \mathbb{N}^{ \times}$, $\mathbb{Z}_{n}$ has exactly $n$ elements

This is exercise 9.2 from textbook Analysis I by Amann/Escher. For $n \in \mathbb{Z}$, $n \mathbb{Z}$ is an ideal of $\mathbb{Z}$, and so the quotient ring $\mathbb{Z}_{n} :=\mathbb{Z} / n \mathbb{...
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1answer
23 views

Multiple definitions of Semirings

I am currently studying for my algebra exam and came across the definition of a semiring. Reading through multiple books at once to better understand the definitions and examples I encountered ...
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3answers
28 views

Hom$_{i[k]}(k[G], R) \approx$ Hom$(G,R^*)$

Let $k$ be a field, $G$ a group and $R$ a $k$-algebra (i.e. a ring $R$ with a homomorphism $i : k \rightarrow Z(R)$). The claim is that there is a natural bijection between the set of $k$-algebra ...
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0answers
25 views

unit ball in $\mathbb C_p$ as an inverse limit

I'm reading Robert's "A Course in $p$-adic Analysis". The Theorem at the bottom of page 79 says that there is an isomorphism $A \cong \varprojlim A/\xi^nA$. ($A$ is the unit ball and $\xi$ is an ...
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2answers
35 views

Why is the unique ring homomorphism $\mathbb Z[x] \to S$, where $x \mapsto s$, irrelevant of $S$ being commutative?

From Aluffi, Algebra: Chapter $0$ If $s$ is any element of a ring $S$, then there is a unique ring homomorphism $\mathbb Z[x] \to S$ sending $x$ to $s$ and ‘extending’ the unique ring homomorphism $...
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1answer
21 views

Display the following vector as a linear combination of the base B.

The base $B\subseteq \mathbb{R}[t]_{\leq 2}$ of the space of polynomials of degree less or equal $2$ is given.$$B=\{2t^2+4t+2,t+3,t^2+5t+3\}$$Display the following vector as a linear combination of ...
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0answers
8 views

For Pairwise Comaximal ideals $I_1, …, I_n$, $I_1 \cap…\cap I_n \subset I_1I_2 \cdots I_n$

In a set of Pairwise Comaximal $I_1, ..., I_n$ ideals of a commutative ring, $I_1 \cap...\cap I_n \subset I_1I_2 \cdots I_n$. I get how to do it for the case when $n=2$: choose $a \in I, b \in J,s.t. ...
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0answers
49 views

What algebraic object do we find at the colimit of $\Bbb Z\to\sqrt 2\cdot\Bbb Z\ldots$? [on hold]

What algebraic object do we find at the colimit of $\Bbb Z\to\sqrt 2\cdot\Bbb Z\ldots$? More precisely (hopefully), consider the function: $g(2^mx)=2^{m/2}x$ Then what is its colimit under infinite ...
1
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1answer
56 views

Questions about $\mathbb C[x]/(x^2+1)$

Given $R$ is the Ring $\frac{\mathbb{C}[x]}{(x^2+1)}$ .Then which of the following option is correct $1. $$R$ has exactly two prime ideal $2.$$R$ is UFD $3.$$(x)$ is a maximal ideal of ...
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0answers
15 views

Determine the characteristic of R[X]/I.

Let I be the ideal in $\mathbb{R}[X]$ generated by $p(X)=X^2+2X+3$. Determine the characteristic of $\mathbb{R}[X]/I$ Am I right in saying the answer is zero as $m(1+I) = m(1)+I \neq 0 $ $\forall m&...
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0answers
32 views

UFD that is not a ED?

Can someone give some examples of a unique factorization domain, that is not a Euclidean domain? I'm aware of $\mathbb{Z}[\frac{1}{2}+\sqrt{-19}]$ and would appreciate any other examples, the simpler ...
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1answer
25 views

Find a group epimorphism $\mathbb{Z}_m \rightarrow \mathbb{Z}_{(m,n)}$ with kernel $n\mathbb{Z}_m$

Let $m$ and $n$ be positive integers and $\otimes=\otimes_\mathbb{Z}$. Denote the GCD of $m$ and $n$ by $(m, n)$. I proved $\mathbb{Z}_m \otimes \mathbb{Z}_n \cong \mathbb{Z}_{(m,n)}$ by considering ...
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1answer
27 views

$(0)$ and $p^n\mathbb{Z}$ (where $p$ prime, $n$ positive integer) are the only primary ideals in $\mathbb{Z}$

I am trying to show that $(0)$ and $p^n \mathbb{Z}$ are the precisely primary ideals in $\mathbb{Z}.$ Clearly $(0)$ is a prime ideal hence primary and radical of $p^n \mathbb{Z}$ being the maximal ...
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2answers
35 views

Multiplication operation on field with four elements whose underlying set is $\mathbb Z/2\mathbb Z \times \mathbb Z/2\mathbb Z$?

Exercise from Aluffi, Alg: Chap. $0$ I know $\mathbb F_4 \cong (\mathbb Z/2\mathbb Z)[x]/(x^2+x+1)$. So, we must have $0 \leftrightarrow (0,0)$, $1 \leftrightarrow (1,1)$ and either $$x \...
1
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1answer
25 views

Prove that $\mathbb{Q}[t] /(f)$ is a principal ideal ring

Let $ f(t)=t^{4}-2 t^{3}+2 t^{2}-2 t+1 $, prove that $\mathbb{Q}[t] /(f)$ is a principal ideal ring. We know that $f(t)$ is the product of two irreducibles: $$t^{4}-2 t^{3}+2 t^{2}-2 t+1 = (t^2+1)(t^...
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1answer
22 views

If $\mathbb Z_{15} / P \cong \mathbb Z_{3}$, then prove that $\mathbb Z{_{15}}_{P} \cong \mathbb Z_{3}$

Here, $\mathbb Z{_{15}}_{P}$ is the localization of the integers by the prime ideal $P$, that is, $\mathbb Z{_{15}}_{P}$=$D^{-1}Z{_{15}}$, for $D = \mathbb Z{_{15}}-P$. I tried this problem by brute ...
0
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1answer
29 views

If $R[x]$ is a PID, $R$ necessarily has to be a field?

It is given that R is a commutative ring with identity. My attempt: I tried to get a contradiction. Given a nonunit $a \in R$, I wanted to show that $(a,x)$ is not a principal ideal in $R[x]$. Then, ...
0
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1answer
26 views

$A$ is a field iff $A[t]$ is euclidean.

I'm almost sure the question has already been asked but i don't know the english terminologies... I have in my lecture that : $A$ a ring. $A$ is a field iff $A[t]$ is principal. I'm anoyed ...
4
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2answers
81 views

Prove $F[x]/(x^n)$ is an injective module

Let $F$ be a field and $n\geq1$ (1) Prove $R=F[x]/(x^n)$ is an injective $R$-module. (2) Give a projective resolution and an injective resolution of the $R$-submodule $M=(x)/(x^n)$ For part (1), I ...
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2answers
38 views

In $\mathbb Z[x]$, is $(2,x)=(2)+(x)$?

The text says that $(2,x)=(2)+(x)$, because $1 \in \mathbb Z$. I do not see why this leads to the decomposition. Can someone point me in the right direction?
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0answers
33 views

Prove that $\mathbb{Z}[i]/\langle a+ib\rangle$ is isomorphic to $\mathbb{Z}/\langle a^2+b^2\rangle$ where $a, b$ are relatively prime [duplicate]

My attempt : Define a map $f$ from $\Bbb{Z}$ to $\Bbb{Z}[i]/\langle a+ib\rangle$ by $f(n) = n+\langle a+ib\rangle$. Then I have shown that $f$ is ring homomorphism and kernel is $\langle a^2+b^2\...
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votes
0answers
17 views

find irreductibles and primes in the ring R={$\frac{a}{b} \in Q$| b is coprime with p,q different primes} [closed]

After finding irreductibles and primes i must have to find the ideals of R and prove that is a PID
0
votes
1answer
53 views

Prime ideals and maximal ideals of the Pullback of rings

Let $A,B,C$ be commutative Noetherian rings with given surjective ring homomorphisms $f:A\twoheadrightarrow C $ and $g: B \twoheadrightarrow C$. Let $A\times_C B:=\{(a,b)\in A \times B : f(a)=g(b)\}$ (...
2
votes
0answers
36 views

Prime Elements and irreducible elements in the ring ${C}[0,1]$

What are the prime elements and the irreducible elements in the ring of real valued continuous functions defined on $[0,1]$? How to find those? I know the prime ideals and maximal ideals in this ...
4
votes
0answers
50 views

When does $v_0\wedge\dots\wedge v_{k-1}=0$ when working over a ring that's not a field?

Let $M$ be a module over a commutative ring $R$, and let $v_0,\dots,v_{k-1}$ be elements of $M$. If $R$ is a field then $v_0\wedge\dots\wedge v_{k-1}$ is equal to $0$ if and only if $v_0,\dots,v_{k-1}$...
0
votes
1answer
23 views

Example of prime ideal [duplicate]

Let $R$ be a nontrivial ring and $P$ be a proper ideal of $R$ such that for any $a, b\in R$ we have $ab \in P$ implies either $a\in P$ or $b\in P$ then $P$ is a prime ideal of $R$. The proof is ...