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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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Reducing an approximation claim from a prime ideal to a maximal ideal

Theorem. Let $R$ be a domain, $K$ its fraction field and $L$ a field extension of $K$. Let $S$ be the integral closure of $R$ in $L$. Let $P_1,...,P_k$ be prime ideals in $S$ with $P_i \cap R = p$. ...
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2answers
50 views

Give an example of a ring with exactly 3 ideals. [on hold]

Give an example of a ring with exactly 3 ideals. (Give a brief explanation.)
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1answer
19 views

Applying degree 2 or 3 irreducibility tests to higher degree

Given the polynomial$\ x^4+x+1$, I have to find out if it is irreducible over $\mathbb Q $. When looking at the solutions, they applied the degree 2 or 3 irreducibly tests to determine that it ...
2
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1answer
34 views

Pre-image of element in quaternion algebra

Let $A$ be an indefinite quaternion algebra (e.g. $(2,5)_\mathbb{Q}$), let $M$ be a maximal order in $A$ and let $\Gamma$ be the Fuchsian group derived from $M$. We will denote the group of units of $...
6
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1answer
54 views

Integral domain which contains a copy of its fraction field

Let $R$ be an integral domain with fraction field $K$. If there exists an injective ring homomorphism from $K$ to $R$ , then is it true that $R$ is a field ? Strictly speaking, I am not saying $K \...
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0answers
15 views

Simple ring that is not semi-simple [duplicate]

Can someone give me an example of a simple ring that is not semi-simple ?
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2answers
29 views

Existence of morphism of rings [on hold]

Given any (possibly non commutative) associative ring with identity $R$, is it always possible to define a morphism of rings from $R$ to $\mathbb{Q}$ that sends $1$ to $1$?
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0answers
11 views

Do we have rings for which $(I:a)=(0:a)$ or for which $(I:a)\cong(0:a)?$

I have been reading a book on Macoy rings and I saw these definitions for the annihilator sets: let $R$ be a ring and $I$ an ideal of $R$. $$(I:a)=\{r\in R: ra\in I\}~~\text{and}~~(0:a)=\{r\in R: ra=...
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1answer
44 views

Computation of completion of a local ring

Let $X=\mathrm{Spec}(\mathbb{R}[a,b]/(a^2+b^2+1))$ and consider the closed point $p=(a)$. I would like to compute the completion of $\mathcal{O}_{X,p}$ w.r.t. to its maximal ideal $\mathfrak m$. ...
2
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1answer
33 views

What is the required homomorphism satisfying $f(c)=c$ for all $c\in R$ and $f(X)=aX+b$?

My question is related to this post. I know from the Proposition that Let $φ : R → R'$ be a ring homomorphism. Given elements $a_1, · · · , a_n ∈ R'$ , there is a unique homomorphism $Φ : R[x_1, ...
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0answers
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Finding the field of fractions of a quotient of a polynomial ring?

This should be very basic but I am having a bit of trouble finding the field of fractions for quotients of polynomial rings over a field. The specific example I am having trouble with is the following:...
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1answer
41 views

Is the trivial ring regular?

In algebraic geometry if $f: X \to Y$ is locally of finite presentation (where $X, Y$ are schemes) then smoothness of $f$ implies that for all $y \in Y$ the "geometric fiber" $\DeclareMathOperator{\...
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0answers
17 views

$R$ UFD and integral domain $\Rightarrow$ every $a\neq 0 \in R\setminus R^\times$ is a product of prime elements

Let $a$ be as in title. Then by definition of UFD we have unique factorisation $$a=\epsilon p_1\dots p_n \quad : \epsilon\in R^\times \text{ and } p_i \in R \text{ irreducible}.$$ It is known that in ...
3
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2answers
54 views

Is $(a)/(a^2)\cong R/(a)$, or something else? ; For ideals $(a), (a^2)\leq R$

(Rings are commutative with 1) In a lecture on introductory commutative algebra, I was presented with an exercise that basically just asked: For a non zero-divisor $a\in R$, we have $(a)/(a^2)\...
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0answers
12 views

Distinct elements in the quotient

How can find the elements in the Ring $\mathbb{Z}_5[x]/\langle (x+1)\cdot(x+3)\rangle$? if it is just $\mathbb{Z}/5\mathbb{Z}$, then it easy to find the elements are $\overline{0},\overline{1},\...
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0answers
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Why the universe of neural network is a ring K?

I wish to find an algebraic or category theory approach to describe neural network in particular to have use algebraic method to extract the concept of 'synaptic weight'. I find this for the moment, ...
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1answer
36 views

How to find the generator of the following ideals?

How to find the generator of the following ideals $\cal a=${$F\in \mathbb Q[X]:F(i)=0$} in $\mathbb Q[x]$, $\cal b=${$F\in \mathbb Q[X]:F(\sqrt 2i)=0$} in $\mathbb Q[x]$? $\cal c=${$F\...
0
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1answer
52 views

For an integral domain $R$, the rings $R\times R $ and $R\times R\times R$ are not isomorphic

For an integral domain $R$, the rings $R\times R $ and $R\times R\times R$ are not isomorphic My attempt: On contrary suppose that both are isomorphic then if G is prime ideal of one ring then its ...
2
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1answer
38 views

Prove a subring of $R=\mathbb{Q}[i]$ is equal to $R$ itself or $\mathbb{Q}$

Consider the ring $R = \mathbb{Q}[i] = \{a + bi \mid a, b ∈ \mathbb{Q}\}$, the subring of $\mathbb{C}$ of all complex numbers with rational real and imaginary parts. Let $T \subset R$ be a ...
3
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1answer
54 views

Show for any simple ring without identity, R, that R is a division ring.

Question: Show for any simple ring without identity, R, that R is a division ring. My thought process is to consider an ideal generated by one element $r$, so I want to consider the ideal $rR$ = $ \...
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1answer
29 views

The localization of an ideal is equal to the localization of the ring

Suppose $m\subset R$ is a maximal ideal. Suppose $I\subset R$ is an ideal. I'm trying to understand these claims: If $m$ does not contain $I$, then $I_m=R_m$ as localizations of $R$-modules. If $m$ ...
3
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2answers
51 views

Find all of homomorphisms $Φ$ from $\mathbb C$[$x$]/$I$ to $\mathbb C$ that satisfies specific condition

I want to know how can I find ALL homomorphism that satisfies the condition mentioned in the question above. Please give me a method or answer in detail.
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0answers
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What is a ring algebra? [duplicate]

Let $R$ be a ring. What does it mean to be an $R$-algebra? I missed the definition in my lectures and when I search online, I can only find definitions of what a ring is.
2
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1answer
37 views

Is it true “If $X \subset Y$ then $\Bbb I(Y) \subset \Bbb I(X)$”(proper inclusion)?

Is it true "If $X \subset Y$ then $\Bbb I(Y) \subset \Bbb I(X)$; here I am using proper inclusion. Couldn't prove it though. Trying for long time please help. Actually I saw here "https://people.maths....
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0answers
49 views

Concerning the Dixmier and the Jacobian Conjectures

I have also asked my following question in MO: Denote by $W$ the first Weyl algebra over a field $K$ of characteristic zero, $W := \langle X,Y | YX-XY=1 \rangle$. Based on Guccione, Guccione, and ...
0
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2answers
63 views

Irreducible polynomial in $\mathbb C[x_1,x_2]$ also irreducible in $\mathbb C[x_1,x_2,…x_r]$? [duplicate]

Let $f_1(x_1), f_2(x_2)$ be polynomials in a single variable, of relatively prime degree, with complex coefficients. If $f_1(x_1)+f_2(x_2)$ is irreducible in $\mathbb C[x_1,x_2]$, then is it ...
-2
votes
2answers
55 views

Idempotents in $ \mathbb{Z}_n $ [on hold]

Let $ n=cd $ where $ c $ and $ d $ are co-primes. Then there are integers $ x $ and $ y $ such that $ xc+dy=1 $. How can it be proved that $ xa $ is idempotent in $ \mathbb{Z}_n $? Is converse ...
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1answer
34 views

Find elements in quotient ring which satisfies specific condition [on hold]

Let $\mathbb { R } [ x ]$ be the polynomial ring in one variable over $\mathbb { R }$ . Let $I$ be the ideal of $\mathbb { R } [ x ]$ generated by the polynomial $x ^ { 3 } - 8 .$ Consider the ...
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2answers
52 views

Why is $(a+b)^p = a^p+b^p$, where $a,b \in R$, a commutative ring with prime characteristic $p$?

Here is the answer from lecture notes. $(a+b)^p = \sum {{p}\choose{k}}a^kb^{p-k}$ and all the terms divide $p$ except $a^p$ and $b^p$ terms. So reducing (mod p) all terms are zero except the ones ...
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2answers
29 views

GCDs for the polynomial ring over a Galois field.

You can find many examples of computing the inverse of an element inside a Galois field. (For example here) What happens if we look at the polynomial ring over a Galois field and would like to ...
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2answers
41 views

Is every ideal in $K[x, y]$ of finite codimension necessarily prime?

I'm trying to answer the following question: Suppose that $R$ is an integral domain containing a field $K$. Then we may view $R$ as a $K$-vector space. Show that if R is finite dimensional as a K-...
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2answers
56 views

Prove a ring isomorphism of stalks

Suppose $f: X \to Y $ is a scheme map with $ f(x) = y $. I want to show that $$ \mathcal{O}_{X, x}/\mathfrak{m}y \, \mathcal{O}_{X,x} \simeq \mathcal{O}_{X_y, x}. $$ Since the question is local, I ...
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1answer
41 views

Solve the equations $x^2= x,~x^2=1$ and $x^{32}=1$ on $\mathbb{Z}_{128}$

On ring $\mathbb{Z}_{128}$, solve each of the following equations: $$(i)~x^2= x,~~~~~~(ii)~x^2=1,~~~~~~(iii)~x^{32}=1.$$ Attempt. Some thoughts. (i) Clearly $x=0,1$ are solutions. Let $x \in \mathbb{...
3
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2answers
94 views

A problem on existence of a zero of a polynomial in a finite field

Let $f(x)$ be a polynomial in $\mathbb Z[X]$ such that degree of $f(x)$ is positive. I want to prove that for infinitely many $p$, $f(x)$ has a zero in $\mathbb{Z}/p\mathbb{\mathbb Z}$. I got a ...
1
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2answers
42 views

Find quantity of elements in group with given order

Let $G = ( \mathbb { Z } / 133 \mathbb { Z } ) ^ { \times }$ be the group of units of the ring $\mathbb { Z } / 133 \mathbb { Z }$ . Find the number of elements of $G$ of order $9 .$ 133 cannot ...
3
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1answer
43 views

How to prove directly sum of non zero divisor and nilpotent is again non zero divisor?

How to prove directly sum of non zero divisor and nilpotent is again non zero divisor? I know that it can be easily proved by extending ring to ring of fraction So that I have a unit as that non zero ...
1
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1answer
38 views

Two orders on projections in $M_2(\mathbb{F}_3)$

Let us consider the space $M_2(\mathbb{F}_3)$ consists of $2\times 2$ matrices over the field $\mathbb{F}_3=\{0,1,2\}$. 1- For a given matrix $A$, we denote its transpose by $A^t$. 2- We say $A$ ...
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2answers
47 views

coprime ideals in $K[X]$

If $K$ is a field, $A=K[X]$, take $m,n \in K$ such that $m \ne n$. Prove that the ideals $I=(X-m)$ and $J=(X-n)$ are coprime. I know the regular definition of coprime. But here, should we prove $I + ...
2
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1answer
26 views

definition of affine polynomial [closed]

I'm reading a paper and it is written inside it "affine polynomial" but I don't know this definition, and couldn't find it on the web. Could you please help me if you know it?
2
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1answer
46 views

On monomial ideals and ring generated by monomials

Question 1: Is $(x^4,x^3y,x^2y^2,xy^3,y^4)$ a maximal ideal in $\mathbb C [ x^4,x^3y,x^2y^2,xy^3,y^4] $? Question 2: Are the ideals $(x^4,x^3y,x^2y^2,xy^3,y^4)$ and $(x^4,x^3y,xy^3,y^4)$ distinct in ...
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2answers
63 views

$x^3 - 3x - 1$ irreducible in $\mathbb Z[x]$ by Gauss Lemma

In Dummit & Foote, they claim this can be shown to be irreducible by Gauss Lemma and applying it to show it has no rational root. But this doesn't make sense to me since Gauss Lemma says: ...
1
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1answer
48 views

Lang's *Algebra*: definition of $F[\alpha]$ and why it's an integral domain?

I am reading Lang's Algebra, namely the chapter about Fields. The first thing which confused me is the following: how he defines $F[\alpha]$? Later he defines this as the smallest subfield of $E$ ...
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0answers
60 views

Hatcher Exercise 3.2.9

Show that if $H_n(X; \mathbb{Z})$ is free for each $n$, then $H^∗(X; \mathbb{Z}_p)$ and $H^∗(X; \mathbb{Z})⊗\mathbb{Z}_p$ are isomorphic as rings. I'm assuming the tensor product is taken over $\...
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2answers
44 views

Irreducible elements of the ring of all numbers of the form $2^ab,$ where $a$ and $b$ are integers

As the title explains, I'm trying to solve a question which asks me to determine which are the irreducible elements of the ring of numbers of the form $2^ab,$ where $a$ and $b$ are integers (with the ...
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0answers
13 views

Let $r\in R$ and let $B$ be any $R$-submodule of a right $R$-module $A$. Then $A/(Ar+B)\cong A/Ar$.

Let $r\in R$ and let $B$ be any $R$-submodule of a right $R$-module $A$. Then $A/(Ar+B)\cong A/Ar$. In the proof, I have defined the map $f:A\to A/Ar$ by $f(a)=a+Ar$ for all $a\in A$. $f$ is well ...
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1answer
25 views

units of polynomial rings [duplicate]

When does a polynomial in the ring of polynomial have an inverse? I thought only constant polynomials were units. if there are other units, under what rings can we guarantee the existence of inverse ...
4
votes
1answer
53 views

When square of an ideal is the square of a maximal ideal in a polynomial ring

Question (1): let $J$ be an ideal in $\mathbb C[X,Y]$ such that $J^2=(X,Y)^2$, then is it true that $J=(X,Y)$ ? Question (2): let $J$ be an ideal in $\mathbb C[X,Y,Z]$ such that $J^2=(X,Y,Z)^2$, then ...
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0answers
20 views

Example of a Jordan homomorphism that is not a homomorphism or antihomomorphism

Can anyone please provide an example of a Jordan homomorphism (preferably on $n\times n$ matrices over a commutative ring) that is not already a homomorphism or antihomomorphism? An obvious Jordan ...
4
votes
1answer
57 views

Show that $M[x] \cong A[x] \otimes_{A} M.$

I'm trying to solve the problems in the book of Atiyah and MacDonald. I want to verify my solution to the problem 2.6. This is the exercise's statement: 2.6. For any $A$-module $M$, let $M[x]$ ...
2
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0answers
68 views

On a necessary condition for being a prime ideal

All rings below are commutative with unity. If $P$ is a prime ideal in a ring $R$, then it has the following property: (*) For every ideal $I,J$ of $R$, $I \cap J \subseteq P \implies I \subseteq P$...