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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1answer
22 views

zero element in tensor product of a localization ring and a module

Let $R$ be a commutative ring with $1$. Let $f$ be a non-nilpotent element of $R$ and let $R_f$ be a localization of $R$ by the multiplicative set $\{ f^i \mid i=0,1,2,\dots\}$. Let $M$ be an ...
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1answer
29 views

If $R$ is a Boolean ring with $\mid R \mid > 2$, determine all the zero divisors of $R$.

If $R$ is a Boolean ring with $\mid R \mid > 2$, determine all the zero divisors of $R$. My attempt: Let $a, b\in R$, $a \neq 0$ and $ab = 0$. How do I prove that $b = 0$?
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2answers
36 views

Show that there exists an idempotent element such that $Ra=Re$ holds for ring $R$

Let $R$ be a ring with 1 such that for every element $x$ in $R$, $\exists y\in R$ such that $xyx=x$ holds. Show that for any $a\in R \exists$ idempotent $e\in R$ such that $Ra=Re$ Let $a=aya$. ...
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2answers
37 views

Intersection of ideals generated by two relatively prime elements

I am wondering how to prove the following statement: Let $R$ be a PID, $a,b$ are relatively prime. Then $\langle a\rangle \cap \langle b\rangle = \langle ab\rangle$ Progress: I think it ...
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0answers
66 views

Finding the left (or right) ideals of the ring of $n\times n$ matrices

Just give me a hint, since this is assessment! DO NOT TELL ME THE IDEAL I want to find the left (or right) ideals of the ring of $n\times n$ complex valued matrices. Now the definition is (for ...
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1answer
34 views

Questions about the formal derivative over $F[x]$

Let $F$ be a commutative ring and $f(x)=a_{0}+a_{1}x+.......+a_{n}x^n$ be in $F[x]$. Define $f'(x)=a_{1}+2a_{2}x+...+na_{n}x^{n-1}$ to be derivative of $f(x)$. Prove that $(f+g)'(x)=f'(x)+g'(x)$, ...
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1answer
32 views

Contradiction in proof that in an integral domain, every prime is irreducible.

Let $\pi$ be a prime element in an integral domain. So, $\pi$ is a non-unit and if $\pi \mid ab \ $ then $\pi \mid a$ or $\pi \mid b$. An irreducible element $z$ is an element such that if $z=ab$, ...
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3answers
49 views

How many elements are in the quotient ring $\frac{\mathbb Z_3[x]}{\langle 2x^3+ x+1\rangle} $

How many elements are in the quotient ring $\displaystyle \frac{\mathbb Z_3[x]}{\langle 2x^3+ x+1\rangle}$ ? I guess I should be using the division algorithm but I'm stuck on how to figure it out.
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2answers
33 views

$\varphi : R → S$ is a epimorphism from $R$ to ring $S$, let $I$ be an ideal of $R$. Prove $\varphi (I) = S$ if and only if $R = I +Ker(\varphi)$

Let $\varphi : R → S$ be an epimorphism from ring $R$ to ring $S$, and let $I$ be an ideal of $R$. Prove that $\varphi (I) = S$ if and only if $R = I +Ker(\varphi)$ I am quite confused on what ...
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1answer
53 views

Find the smallest subring of $\mathbb{R}$ containing $\frac 12$.

Find the smallest subring of $\mathbb{R}$ containing $\frac 12$. My attempt: I have formed a subring containing $\frac 12$ i.e. $\{\frac n2 | n \in \mathbb{Z}\}\cup\{(\frac 12)^{k}|k\in\mathbb{Z}, ...
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2answers
54 views

Find the smallest subring of $\mathbb{Z}$ containing $8$.

Find the smallest subring of $\mathbb{Z}$ containing $8$. My attempt: I have formed a subring containing 8 i.e. $\{8n \mid n \in \mathbb{Z}\}\cup\{8^k \mid k\in\mathbb{Z}, k>0\}$. But how do I know ...
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2answers
58 views

What is an example of a homomorphism of rings that doesn't preserve gcd's?

Given a commutative ring $R$, we say that $x$ is a gcd of $(y,z)$ iff the following conditions hold: $x \mid y,z$ For all $x' \in R$, if $x' \mid y,z$, then $x' \mid x$. This gives a ternary ...
7
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2answers
109 views

$1+\frac{1}{2} +\frac{1}{3} +…+\frac{1}{p-1} =\frac{a}{b}$

Let $p\gt 3$, be a prime number and $1+\frac{1}{2} +\frac{1}{3} +...+\frac{1}{p-1} =\frac{a}{b}$ when $a,b\in \mathbb N$ and $gcd(a,b)=1$. prove that $p^2|a$. I proved that $p|a$, but I cant ...
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1answer
75 views

How is Z$/n$Z isomorphic to Z$_n$?

Let $n$Z be the set of integer multiples of $n \in$ Z. Can someone explain how Z$/n$Z is isomorphic to Z$_n$? Specifically, what is the function that establishes the isomorphism and how can we be ...
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5answers
134 views

In a ring with no zero-divisors, for $(m,n) =1$, $a^m = b^m$ and $a^n = b^n$ $\iff a =b$

Let $R$ be a ring with with no zero divisors. If $a, b \in R$ are such that $a^m = b^m$ and $a^n = b^n$, where $m$ and $n$ are relatively prime positive integers, then show that $a = b$. My ...
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1answer
29 views

Why is the degree condition for a degree reverse lexicographic order necessary?

A degree reverse lexicographic order $\prec$ is defined as follows: Given the polynomial ring $R=K[x_1,...,x_n]$. Two monomials in $R$ have the order $x^u\prec x^v$, if $\deg(x^u)<\deg(x^v)$, or ...
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4answers
420 views

Definition of prime element in a Euclidean ring does not make sense. Herstein - Topics in Algebra

Herstein's Definition: In the Euclidean ring $R$, a nonunit $\pi$ is said to be a prime element of $R$ if whenever $\pi=ab$, where $a,b \in R$, then one of $a$ or $b$ is a unit in R. $\mathbb Q$ is a ...
2
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1answer
34 views

Show that $P = (f(x))$ is a maximal ideal of $F(x)$

Let $F$ be a field and $f(x)$ be an irreducible polynomial in $F[x]$. Prove that $P = (f(x))$ is maximal in $F[x]$. (Here is what I know: $f(x) \neq 0 \wedge f(x) \not \in U(F[x])$, since $f(x)$ is ...
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0answers
24 views

Prove that $a$ is a prime element of $R$

Let $R$ be a PID and $P = (a)$ is a prime ideal of $R$. Prove that $a$ is a prime element of $R$. Since $P$ is a prime ideal of $R$, let $x,y \in R$ s.t. $xy \in P = (a).$ (WTS $a \mid x$ or $a\mid ...
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1answer
25 views

Ring Homomorphism Textbook Question

Please help me understand the last three sentences in this paragraph from the Artin textbook. Where does this come from: "The monomials that appear in $r_0(t^2)$ have even degree, while those in ...
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2answers
30 views

Ring Homomorphism Defintion

$\varphi:R\rightarrow S$ is said to be a ring homomorphism if, $R,S$ are rings and $\varphi$ is a map such that: $\varphi(r_{1}+r_{2})= \varphi(r_{1})+\varphi(r_{2})$, $\varphi(r_{1}.r_{2})= ...
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2answers
41 views

A given ring of matrices has an infinite number of invertible elements

The set $\mathcal{M} = \bigg\{ \begin{pmatrix} a & 2b \\ b & a \\ \end{pmatrix} \bigg\vert a,b \in \mathbb{Z} \bigg\}$ is given. Prove that: (1) $\mathcal{M}$ is a commutative ring with ...
2
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1answer
45 views

Least common multiple for integer matrices

Given two full-rank $3\times3$ integer matrices $M_1$ and $M_2$, I am trying to find integer matrices $N_1$ and $N_2$ such that $M_1N_1$=$M_2N_2$, such that $\left|\det(M_1N_1)\right|$ is minimal. ...
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1answer
61 views

How to define the isomorphism?

Let $R$ be a ring, then For $R[x]/\langle x-1\rangle \cong R$, we define the map, $\varphi$ : $R[x]\rightarrow R$, defined by $\varphi(f) =f(1)$ For $R[x]/\langle x\rangle \cong R$, we define the ...
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1answer
28 views

Boolean algebras and rings

I know that M. H. Stone proved that there is a bijection between boolean algebras and boolean rings. The bijection I know is the following: to any given Boolen algebra $(L,\, \vee, \wedge)$ we ...
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0answers
33 views

ideals of polynomial ring with complex number coefficients

Let $\mathbb{C}[x,y]$ be the polynomial ring with variables $x,y$ and coefficient in $\mathbb{C}$. Let $f,g\in \mathbb{C}[x,y]$. Let $(f,g)$ be the ideal of $\mathbb{C}[x,y]$ generated by $f,g$. ...
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1answer
20 views

Ring polynomial kernel generators

This is the textbook question: Q: Find generators for the kernels of the following maps: $\mathbb{R}[x,y] \to \mathbb{R}$ defined by $f(x,y) \mapsto f(0,0)$ $\mathbb{R}[x] \to \mathbb{C}$ defined ...
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1answer
32 views

If $P$ is a prime then $R/P$ is an integral domain.

I know the same question has been already asked here. So, I am not asking for any proof rather to find out what's wrong with my proof. So, this is what I did: Let, $a+p, b+p \in R/P$, since $P$ is a ...
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1answer
37 views

Show that it is a homomorphism?

For any abelian group $G$ we have $e_n: G \to G, e_n(g) = g^n$. By convention $e_0(g) = 1$. For a Field $F$ we have the subgroup $\{1,-1\} \leq F^*$. When $F$ is of characteristic $2$, this is the ...
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2answers
61 views

What Notation is this?

When $p$ is prime, show that $v: Z^*_p \rightarrow U_2$ I know that the $Z_p$ is the elements $\{0,1,2,\cdots,p-1\}$ But what about the star on top of the $p$? Is that the group operation? Because ...
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2answers
69 views

Give an intuitive explanation for polynomial quotient ring, or polynomial ring mod kernel

I learned how to see quotient groups intuitively when I learned of a group mod its commutator subgroup. If we take a group and mod out all the elements that do not commute, we get a quotient group ...
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1answer
31 views

Find the Number of Elements of a Particular Quotient Ring

Find the size of $\mathbb{Z}[\sqrt{-19}]/I$, where $I=(18+\sqrt{-19}, 7)$. The standard way to proceed would be $\mathbb{Z}[\sqrt{-19}]/I=\mathbb{Z}[x]/(x^2+19, 18+x, 7)=\mathbb{Z}_7[x]/(x^2+5, ...
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1answer
32 views

How to find all roots of the equation $ x^3 + 2x^2 - 3x$ in $\mathbb Z_{12}$

Firstly you can factor it completely from $ x^3 + 2x^2 -3x$, which is $x(x-3)(x+1)$. We have the obvious roots of $0$, $3$ and $-1$, but what about the other roots? I have a little confusion here ...
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1answer
21 views

Is $T_n(R) \cong T_n(R)^{op}$?

I am working on the following problem: Let $R$ be a commutative ring, and $T_n(R)$ be the ring of $n \times n$ upper triangular matrices. Is $T_n(R) \cong T_n(R)^{op}$? I have already shown ...
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0answers
40 views

Krull's height theorem in the non-Noetherian case

Krull's height theorem says that if $R$ is a Noetherian ring and $I$ is a proper ideal generated by $n$ elements of $R$, then $\operatorname{ht} I\le n$. When $R$ is not Noetherian, this is not ...
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2answers
36 views

Show that $B/Q$ is integral over $A/P$

If $A$ is a subring of $B$ and $B$ is integral over $A$, let $Q$ be a prime ideal of $B$ and $P=Q\cap A$. Show that $B/Q$ is integral over $A/P$. If $b\in B$ is integral over $A$ then for some ...
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2answers
20 views

Simple generator modules

Let a ring $R$ with identity element be such that the category of left $R$-modules has a simple generator $T$. My question: "Is $T$ isomorphic with any simple left $R$-module $M$?" I tried the ...
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0answers
33 views

Question about the Chinese Remainder Theorem and the residue class ring ${\bf Z}/p\# {\bf Z}$

In a question that I asked on MO, Terence Tao observed that: The Chinese Remainder Theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of ...
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1answer
66 views

Why do we have a basis?

A corollary that is in my book that I think is relevant to my question is: If E is an extension field of F, $\alpha \in E$ is algebraic over F, and $\beta \in F(\alpha)$, then $\deg(\beta,F)$ ...
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0answers
33 views

Need Help Understanding Why Proof Shows Set is not a Ring

I am having trouble reading this somewhat "slick" proof. Maybe it's not as slick as I think it is though, and I'm missing something here. So, I understand everything that is being done until the last ...
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0answers
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$x\in\mathcal{O}_K$ can be written as product of irreducible elements

Lemma: Every element $x\neq 0$ of $\mathcal{O}_K$ with ($K$ an arbitrary number field) can be written as a product of irreducible elements. Proof: We prove this lemma by complete induction on ...
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1answer
32 views

What is the difference between these two conditions $J = \{az \mid a \in R\}$ and $ I = \{a \in R \mid az \in J\}$

Please consider these two questions: Let $R$ be a ring and $z \in R$, which is fixed. Let, $J = \{az \mid a \in R\}$. Prove that $J$ is a left ideal of $R$. Skipping the subtraction part, this is ...
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3answers
31 views

Why is it the smallest subfield containing F and $\alpha$?

Please take a look at the sentence in red: I understand that $\phi_\alpha[F[x]]$, is a subfield which contains $\alpha$, and F(we just need to evaluate $\phi_\alpha$ at the appropriate values). But ...
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1answer
21 views

The annihilator induces a module [duplicate]

Let $R$ be a ring, and $M$ an $R$-leftmodule. Let $\operatorname{Ann}_R(M)$ be the annihilator of M, meaning that $r m = 0 \space\space\space\space \forall r \in \operatorname{Ann}_R(M), m \in M$. ...
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0answers
29 views

Question about the deduction of the quotient ring $R/I$

Yesterday we deduced on class how quotient groups were deduced and well defined. Let $R$ be a ring and $I$ an ideal of $R$. My professor proved us that the multiplication operation $$R/I \times R/I ...
3
votes
3answers
54 views

Prove or disprove $\mathbb{Q}[x] /(x^5-3) \cong \mathbb{Q}[x] /(x^5-9)$

Want to prove or disprove this $\mathbb{Q}[x] /(x^5-3) \cong \mathbb{Q}[x] /(x^5-9)$ as communtative rings. I can show that $x^5-3$ and $x^5-9$ are irreducible in $\mathbb{Q}$, but I cannot go from ...
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0answers
12 views

Show that if $R$ and $S$ are ideal of a ring $A$ then the product $R\cdot S$ is a ideal of $A$. [duplicate]

How to prove that if $R$ and $S$ are ideal of a ring $A$ then the product $R\cdot S$ is a ideal. I can't show only that if $x, y\in R\cdot S$ then $x-y\in R\cdot S$. The other axioms of ideal I ...
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3answers
58 views

Example of ideal generated by two elements

I have an easy example on my notes that I don't understand. My teacher said that in $\mathbb{Z}$, $(2,3)=2\mathbb{Z}+3\mathbb{Z}$ is a principal ideal, because $2\mathbb{Z}+3\mathbb{Z}=\mathbb{Z}$. ...
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1answer
18 views

Is group of units of a polynomial ring only constant polynomial which is involved in R

Let R be a integral domain(or maybe field) edit : Let R be a field. The group of units of R[x] is $$ a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots+ a_2 x^2 + a_1 x + a_0 $$(or infinity) such ...
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0answers
19 views

$x^{mn} -a$ is irreducible in F[x] iff $x^m -a$ and $x^n -a$ are irreducible.

Let F be any field, a is in F and (m,n)=1. Show that $x^{mn}-a$ is irreducible in F[x] iff $x^m -a$ and $x^n -a$ are irreducible in F[x]?