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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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0answers
14 views

Can I make the correspondence theorem for rings similar to that of the one for groups?

I tried to follow the correspondence theorem for groups to write out the correspondence theorem for rings. Let $I$ be an ideal in a ring $R$. Then there exists a one-to-one correspondence (a ...
1
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3answers
38 views

Finding the minimal polynomial of $2+\sqrt3$ over $\mathbb{Q}$

Minimal polynomial of $2+\sqrt3$ over $\mathbb{Q}$. $(2+\sqrt3)^2$= $7+4\sqrt3$ $\implies$ $((2+\sqrt3)^2-7)^2$=$48$ $\implies$ $((2+\sqrt3)^4-14(2+\sqrt3)^2+1=0$ So $2+\sqrt3$ is a root of $x^4-...
2
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0answers
45 views

Finding $(t^3+3)^{-1}$ if $t^5+2t+2=0$

Finding $(t^3+3)^{-1}$ if $t^5+2t+2=0$. I started trying to do this one by the euclidean algorithm but I wasn't able to get the remainder down to a unit. Anyone got any tips?
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1answer
23 views

Problems about sum of ideals

Let $I_1,I_2,\dots ,I_n$ be ideals of the ring $R$ with $R=I_1+I_2+\cdot\cdot\cdot +I_n$. Show that this sum is direct if and only if $a_1+a_2+\cdot\cdot\cdot+a_n=0$, with $a_i\in I_i$, implies that ...
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2answers
23 views

Prove that $\frac{\mathbb{F}_2[x]}{(x^2+1)} \cong \frac{\mathbb{F}_2[x]}{(x^2)}$

I need to show that $\frac{\mathbb{F}_2[x]}{(x^2+1)}$ is isomorphic to $\frac{\mathbb{F}_2[x]}{(x^2)}$. However, I am becoming increasingly convinced that they are not. I believe the members of both ...
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0answers
21 views

how to prove a graded version of Nakayama's lemma

There is a simple version of graded Nakayama lemma in Wikipedia:https://en.wikipedia.org/wiki/Nakayama%27s_lemma :If $R$ is a positive graded ring,Let $M$ be a graded $R$ module such that $M_i=0$ for $...
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0answers
23 views

Ring with infinitely reducible elements

Can you give or construct an elementary example of a factorial ring with elements which are product of infinitely many irreducible elements? i.e. there are reducible elements that can't be written as ...
4
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2answers
59 views

Find the minimal polynomial of $t^2+t$ over $\mathbb{Q}$ where t satisfies $x^3-3x^2-3$

Find the minimal polynomial of $t^2+t$ over $\mathbb{Q}$ where t satisfies $t^3-3t^2-3=0$. Okay, so I was working on this one for awhile today with my buddy and we couldn't figure it out, haha. We ...
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0answers
14 views

Artinian integral domains are fields. [duplicate]

I was thinking about the question above and can't find an easy way of proving it. Any suggestions?
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1answer
31 views

Problems about the ideal generated by a set

Let $I$ be an ideal of $R$, a commutative ring with identity. For an element $a\in R$, the ideal generated by the set $I\cup\{a\}$ is denoted by $(I,a)$. Assuming that $a\notin I$, it can be shown ...
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1answer
52 views

Let $ R = \mathbb{ F }_2[x]/(x^2) $. Determine all the ideals in $ R $. [on hold]

(3) Let $ R = \mathbb{ F }_2[x]/(x^2) $. Determine all the ideals in $ R $. $$ R := \frac{ \mathbb{ F }_2[x] }{ (x^2) },$$ the quotient of the polynomial ring $ \mathbb{ F }_2[x] $ (read "f ...
3
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2answers
20 views

The lattice of annihilator ideals of a ring

The question is about an exercise from the book "Lattice-ordered rings and modules" from Stuart A. Steingberg. This is the exercise 7 from chapter 1, section 2. Let $R$ a ring with no nonzero ...
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0answers
24 views

Subdomains of matrix algebras

Let $F$ be a field and M$_n(F)$ the ring of $n\times n$ matrices. By a domain we mean a not necessarily commutative ring without zero divisors. We consider subdomains $R$ of the ring M$_n(F)$. ...
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1answer
16 views

Distributive lattices

I have a question which is in my ring theory lesson. it's under the topic of distributive lattice and I don't know how to prove it. Que: If A is a strongly regular ring, then the principle right ...
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0answers
20 views

Are the following definitions of a local property equivalent?

I have seen two different definitions of a local property of rings. $P$ is a local property of rings if $P(A)$ is equivalent to $P(A_{\mathfrak p})$ for all prime ideals $\mathfrak p$. $P$ is a local ...
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1answer
18 views

Relationship between $\mathcal{Spec}(R)$ and $\mathcal{Spec}(R_{\text{red}})$

Let $R$ be a commutative ring. I am wondering if there is any well-known relationship between $\mathcal{Spec}(R)$ and $\mathcal{Spec}(R_{\text{red}})$ that would allow one to conclude $[\mathcal{...
2
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1answer
32 views

$A$ Noetherian and $f:A \rightarrow A$ suryective then $f$ inyective [duplicate]

I think this must have been questioned before, but after searching, I couldn't find it. I thought of considering a set of $\{x_1, ..., x_n\}$ such that $\lt x_1, ... x_n\gt = A$. The hypotesis shows ...
7
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2answers
279 views

The group ring of a ring.

Let $R$ be a ring. Since $R$ is also a group then we can talk about the group ring $R[R]$. I want to understand this group ring $R[R]$. An element $x\in R[R]$ is written as a finite formal sum $$x=...
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0answers
31 views

Every left ideal of $R[x]$ is cyclic implies $R$ a division ring

I need to prove that if every left ideal of $R[x]$ is cyclic as a left $R[x]$-module then $R$ is a division ring. I don't really understand. So a left ideal $I$ is cyclic as a left $R[x]$-module, let ...
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2answers
50 views

Example of integral domain with “non-trivial” units

The units of $\mathbb{Z}$ are 1 and -1, this is quite "easy" to see. Are there any rings $R$ which has "non-trivial" units, in the sense that it takes some work to figure out what are the units of $R$...
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1answer
46 views

Ring Homomorphism from $\mathbb{Z}$ to a Field $F$ [duplicate]

I have so far shown that $\phi: \mathbb{Z} \to F$ is a unique ring homomorphism if $\phi(n)=n$ and that $ker(\phi) = (n) = \{nm|m \in \mathbb{Z}\}$ with $n = 0$ or $n=p$ with $p$ a prime number. ($n$ ...
1
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2answers
40 views

Unique factorization conjecture?

Let $A_p$ be an Integral domain. Conjecture : If every $a$ in $A_p$ that equals $b \space c$ for irreducible elements $b,c$ in $A_p$ , has Unique factorization then the Integral domain $A_p$ is a ...
0
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1answer
54 views

Proving that Matrix is a Unit [duplicate]

If $A \in M_{n}(F)$, I have to show that $A$ is a unit only if $AB = I$ or $BA = I$ for some $B \in M_{n}(F)$. I am not sure how to approach this at all since this fact was pretty intuitive to me. ...
1
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1answer
23 views

Set of Units of Continuous functions

I am asked to find set of units, $R^{\times}$, of a ring of continuous functions $\mathbb{R} \to \mathbb{R}$ denoted as $C(\mathbb{R})$. Now, unit is $a \in R$ such that $ab = 1$ for some $b \in R$. ...
2
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2answers
49 views

Let $\theta$ be a root of $p(x)=x^3+9x+6$, find the inverse of $1+\theta$ in $\mathbb{Q(\theta)}$

Let $\theta$ be a root of $p(x)=x^3+9x+6$, find the inverse of $1+\theta$ in $\mathbb{Q(\theta)}$. So problems like this really annoy me but I did crappy on the last homework after making a lot of ...
0
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0answers
8 views

The left ideals of $R$ admit decompositions as sums left ideals whenever $R$ admits a decomposition as a sum of ideals [duplicate]

Let $R=B_1 \oplus \dots B_n$ where the $B_i$ are ideals of $R$. Then, there are idempotents $e_i \in B_i$ such that the $B_i$ are rings with identity $e_i$ and for $i \neq j$ $e_ie_j=0$. We wish to ...
1
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2answers
30 views

Gcd of two elements

Consider the ring $\mathbb{Z}[\sqrt{2}]$. I need to find $\gcd(4, 6)$. My try Let $N$ be norm function defined on $\mathbb{Z}[\sqrt{2}]$ and $d$ be proper divisor of $4$ and $6$ then $d$ can't be ...
4
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1answer
44 views

Adjoining $1$ to non-unital ring: is the usual way the best way?

In Modern Higher Algebra written by A. Adrian Albert (1938), the characteristic of a non-unital ring $R$ is defined the least positive integer $m$ such that $ma=0$ for all $a\in\mathbb R$. If such an ...
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0answers
23 views

A[X]/(aX+b) is isomorphic to A_a [on hold]

I don't know exactly the result but I know that si some think like this. $A$ domain, $a,b\in A$ then $A[X]/(aX+b)$ is isomorphic to $A_a$ ($A$ localized at $a$). I can't remember the exact statment ...
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2answers
49 views

Rings isomorphic to a proper subring

Is there a theory for rings which are isomorphic to a proper subring? Which of the following rings have this property? $$ \mathbb{R} , M_2(\mathbb{R}) , \mathbb{C} \; and \; M_2(\mathbb{Z})$$
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1answer
56 views

In the ring $\Bbb Z_6$ commutative ring with $1$,why $[3]$ is not irreducible?

Though I was thinking in that way. $[3]=[1]*[3]$, where $[1]$ is unit of $\Bbb Z_6$ and $[3]$ is nonzero nonunit, so why not $[3]$ is irreducible in $\Bbb Z_6$. Please explain.
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0answers
20 views

If $R$ is a ring such that $x^2=x$ $\forall x \in R$ and $I$ is a prime ideal. Show that $R/I$ has two elements

If $R$ is a ring such that $x^2=x$ $\forall x \in R$ and $I$ is a prime ideal. Show that $R/I$ has two elements. $R/I = \{ r+I:r\in R \}$ Let $a \in R$ if $a\in I$ then $a+I = 0+I=I$. if $a\notin ...
4
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2answers
49 views

Is my solution correct on this Groups/Rings

So basically, I'm still learning about Groups/Rings and I was wondering if my solution for this exercise is correct. Also how can I solve the b part of the exercise (since i'm not sure). Exercise: On ...
1
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2answers
61 views

Expressiong $\frac{t+2}{t^3+3}$ in the form $a_o+a_1t+…+a_4t^4$, where $t$ is a root of $x^5+2x+2$

Expressing $\frac{t+2}{t^3+3}$ in the form $a_o+a_1t+...+a_4t^4$, where $t$ is a root of $x^5+2x+2$. So i can deal with the numerator, but how do I get rid of the denomiator to get it into the ...
2
votes
1answer
29 views

Use induction to prove that $(a+b)^{p^n}=a^{p^n}+b^{p^n}$

Use induction to prove that $(a+b)^{p^n}=a^{p^n}+b^{p^n}$ where $a,b \in F$ and $char(F)=p$. So, i'm having problems doing this like a normal induction problem because i'm doing induction on the ...
0
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1answer
17 views

In a ring with identity, prove that (−1) · (−1)= 1. [duplicate]

I'm not sure how to start this problem with Rings, could someone perhaps lead me in the right direction?
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1answer
25 views

Let $\alpha: F \rightarrow F$ be a one to one field homomorphism, prove $\alpha$ fixes the prime subfield of $F$ elementwise

Let $\alpha: F \rightarrow F$ be a one to one field homomorphism, prove $\alpha$ fixes the prime subfield of $F$ elementwise. So I already proved that $\alpha$ fixes the identity element and used ...
1
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1answer
31 views

Division on a real quadratic ring of integers.

I've seen that $\mathbb{Z}[\varphi ] = \mathcal{O}_{\mathbb{Q}_{\sqrt{5}}}$, where $\varphi$ is the golden ratio, is a Euclidean domain with norm $N(x + y\varphi ) = x^{2} + xy - y^{2}$. Given a ...
2
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1answer
24 views

Sum, difference and product of algebraic elements is an algebraic element.

I found the proof of the result "the Sum, difference and product of algebraic elements of a ring $S$ over a subring $R$, is an algebraic element over $R$", but I failed to find a polynomial in the ...
2
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1answer
70 views

Given $a, b$ coprime integers, show that any factor of $a^2 - 2b^2$ is of the form $c^2 - 2d^2$

This is an exercise from Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory. I imagine the relevant ring here is $Z[\sqrt{2}]$, i.e., we can factor $a^2 - 2b^2$ into $(a + b\...
2
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1answer
26 views

Rings with decomposition are sums of ideals generated by idempotents [duplicate]

I'm working on a question out of T.Y. Lam's book that has me thrown. Let $B_1 \dots, B_n$ be left ideals (resp. ideals) in a ring $R$. Show that $R=B_1 \oplus \dots \oplus B_n$ iff there exists ...
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0answers
18 views

$R$ a $PID \Rightarrow R[[X]]$ a $UFD$ [duplicate]

I have just shown that an integral domain $R$ is a unique factorization domain iff for every non-zero prime ideal $P$ of $R$, $P$ contains a non-zero prime, principal ideal. I am then asked to show ...
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0answers
19 views

Semisimple matrix ring [duplicate]

Let $R$ be a ring and suppose the matrix ring $M_n(R)$ is semisimple. How does one proof that $R$ is semisimple?
2
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3answers
41 views

Show that every maximal ideal of $R$ contains the element $a$.

Let $R$ be a commutative ring with identity and let $a\in R$ such that $a^n=0$, for some positive integer $n$. Suppose that $I$ is and ideal of $R$. Define $(I,a)=\{x+ra\ :\ x\in I\ \text{and}\ r\in R\...
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1answer
25 views

$R$ is a ring with $\operatorname{char}(R)=mn$ where $(m,n)=1$, then there exists an ideal $A$ such that $\operatorname{char}(A)=m$?

whole question : Let $R$ be a ring with characteristic $mn$ for some positive integer $m,n$. If $(m,n)=1$, show that there exists an ideal $A$ (resp. $B$) of $R$ with characteristic $m$ (resp. $n$). ...
6
votes
4answers
43 views

If $r$ is a nilpotent element of a commutative ring $R$ then $(r)$ is not a direct summand of $R$ as an $R$-module.

I am having trouble proving that if $r$ is a nilpotent element of a commutative ring $R$ then $(r)$ is not a direct summand of $R$ as an $R$-module. I know that $(r)$ is then a nilpotent ideal. I ...
3
votes
2answers
49 views

$(1-ba)$ left-invertible $\implies (1-ab)$ left-invertible

Suppose $(1-ba)$ in a ring $R$ is left-invertible. Then we wish to show that $(1-ba)$ is left invertible and explicitly construct its inverse. We have $Rb(1-ab)=R(1-ba)b=Rb \subseteq R(1-ab)$. I'm ...
3
votes
1answer
50 views

Action of sum of traspositions on a simple $\mathbb{C}[S_n]$-module

Let $\lambda$ be a Young tableaux and $V_{\lambda}$ the standard simple $\mathbb{C}[S_n]$-module associated to it, constructed as $\mathbb{C}[S_n]c_{\lambda}$. We denote by $x=\sum\limits_{1\leq i &...
0
votes
0answers
40 views

Prove that $\left(\forall\ a \in A \right)\ aTe = eTa = e$

Let $\left(A,\ *, T\right)$ be a Ring and $e$ be its identity element. Is it possible to prove the following statement without using the regularity of $aTe$ in the underlying group $\left(A,\ *\right)...
1
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0answers
54 views

Looking for a pair of adjoint functors

I'm struggling with finding the left adjoint to a functor, and the right adjoint to another one. Here's some context. Given any ring $R$, we can associate two categories to it. The first one is a ...