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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Units and Nilpotents

If $ua = au$, where $u$ is a unit and $a$ is a nilpotent, show that $u+a$ is a unit. I've been working on this problem for an hour that I tried to construct an element $x \in R$ such that $x(u+a) ...
9
votes
3answers
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Characterizing units in polynomial rings

I am trying to prove a result, for which I have got one part, but I am not able to get the converse part. Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots ...
33
votes
6answers
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Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong ...
26
votes
2answers
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Why is $\mathbb{Z}[\sqrt{-n}]$ not a UFD?

I'm considering the ring $\mathbb{Z}[\sqrt{-n}]$, where $n\ge 3$ and square free. I want to see why it's not a UFD. I defined a norm for the ring by $|a+b\sqrt{-n}|=a^2+nb^2$. Using this I was ...
12
votes
4answers
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Zero divisor in $R[x]$

Let $R$ be commutative ring with no (nonzero) nilpotents. If $f(x) = a_0+a_1x+\cdots+a_nx^n$ is a zero divisor in $R[x]$, how do I show there's an element $b \ne 0$ in $R$ such that ...
11
votes
4answers
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A ring is a field iff the only ideals are $(0)$ and $(1)$

Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$. I can't figure out where to start other that I ...
16
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8answers
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How to show that every Boolean ring is commutative?

A ring $R$ is a Boolean ring provided that $a^2=a$ for every $a \in R$. How can we show that every Boolean ring is commutative? Thanks in advance.
29
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5answers
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Why can't the Polynomial Ring be a Field?

I'm currently studying Polynomial Rings, but I can't figure out why they are Rings, not Fields. In the definition of a Field, a Set builds a Commutative Group with Addition and Multiplication. This ...
16
votes
2answers
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A subring of the field of fractions of a PID is a PID as well.

Let $A$ be a PID and $R$ a ring such that $A\subset R \subset \operatorname{Frac}(A)$, where $\operatorname{Frac}(A)$ denotes the field of fractions of $A$. How to show $R$ is also a PID? Any ...
18
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5answers
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Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
32
votes
1answer
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An example of a division ring $D$ that is **not** isomorphic to its opposite ring

I recall reading in an abstract algebra text two years ago (when I had the pleasure to learn this beautiful subject) that there exists a division ring $D$ that is not isomorphic to its opposite ring. ...
22
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7answers
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Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$

Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please. The ring that I am talking about is ...
12
votes
4answers
4k views

Why are maximal ideals prime?

Could anyone explain to me why maximal ideals are prime? I'm approaching it like this, let $R$ be a commutative ring with $1$ and $A$ be a maximal ideal. Let $a,b\in R:ab\in A$ I'm trying to ...
7
votes
2answers
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What are the left and right ideals of matrix ring? How about the two sided ideals?

What are the left and right ideals of matrix ring? How about the two sided ideals?
12
votes
2answers
1k views

$A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$

I'm trying to prove that if $A\neq 0$ is a commutative ring and there is an injective $A$-module homomorphism $A^m\hookrightarrow A^n$ then $m\leq n$ must necessarily hold. This is exercise 2.11 ...
0
votes
1answer
227 views

Prove $R \times R$ is NOT an integral domain [duplicate]

I have a question, and this is it in entirety: Let R be a non-zero ring(that is, R contains at least one element other than the zero element). Prove that $R \times R$ is NOT an integral domain. ...
23
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3answers
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Why doesn't $0$ being a prime ideal in $\mathbb Z$ imply that $0$ is a prime number?

I know that $1$ is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=0$ is ...
3
votes
2answers
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Why can we use the division algorithm here?

In Theorem 5.2.3 in these notes, it is said that Since $(x − a)$ has leading coefficient $1$, which is a unit, we may use the Division Algorithm... Why is this true? I thought that the ...
13
votes
3answers
3k views

A ring that is not an Euclidean domain

Let $\alpha = \frac{1+\sqrt{-19}}{2}$. Let $A = \mathbb Z[\alpha]$. Let's assume that we know that its invertibles are $\{1,-1\}$. During an exercise we proved that: Lemma: If $(D,g)$ is an ...
33
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3answers
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Does every Abelian group admit a ring structure?

Given some Abelian group $(G, +)$, does there always exist a binary operation $*$ such that $(G, +, *)$ is a ring? That is, $*$ is associative and distributive: \begin{align*} &a * (b * c) = ...
11
votes
3answers
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The subring test

This is how the wikipedia article on subring defines the subring test The subring test states that for any ring $R$, a nonempty subset of $R$ is a subring if it is closed under addition and ...
2
votes
1answer
630 views

The number of ring homomorphism from $\mathbb{Z_m}$ to $\mathbb{Z_n}$

I face the problem solving how many non trivial ring or group homomorphism from $\mathbb Z_m$ to $\mathbb Z_n$ ($m<n$), is there any general formula? At this moment, I want to know how many ring ...
16
votes
6answers
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Why is a finite integral domain always field?

This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse: let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$, then as $R$ is closed under ...
34
votes
2answers
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Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?

Why are the only (associative) division algebras over the real numbers the real numbers, the complex numbers, and the quaternions? Here a division algebra is an associative algebra where every ...
5
votes
3answers
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Ring of trigonometric functions with real coefficients

Let $R$ be the ring of functions that are polynomials in $\cos t$ and $\sin t$ with real coefficients. Prove that $R$ is isomorphic to $\mathbb R[x,y]/(x^2+y^2-1)$. Prove that $R$ is not a unique ...
4
votes
1answer
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Submodule of free module over a p.i.d. is free even when the module is not finitely generated?

I have heard that any submodule of a free module over a p.i.d. is free. I can prove this for finitely generated modules over a p.i.d. But the proof involves induction on the number of generators, so ...
11
votes
5answers
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Proving that surjective endomorphisms of Noetherian modules are isomorphisms and a semi-simple and noetherian module is artinian.

I am revising for my Rings and Modules exam and am stuck on the following two questions: $1.$ Let $M$ be a noetherian module and $ \ f : M \rightarrow M \ $ a surjective homomorphism. Show that $f ...
10
votes
2answers
840 views

Find all subrings Of $\mathbb{Z}^2$

This may be a simple question: Find all subrings of $\mathbb{Z}^2$.
4
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2answers
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In a finite ring extension there are only finitely many prime ideals lying over a given prime ideal [duplicate]

I'm trying to solve the exercise 6.7 of Miles Reid's Undergraduate Commutative Algebra (pag 93). How can I prove that if $B$ is a finite ring extension of $A$, there are only finitely many prime ...
4
votes
3answers
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Left inverse implies right inverse in a finite ring

Let $R$ be a finite ring, and assume $\exists x,y\in R$ such that $ xy=1$. How can I show it implies $yx=1$?
19
votes
2answers
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necessary and sufficient condition for trivial kernel of a matrix over a commutative ring

In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ...
18
votes
5answers
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Why is ring addition commutative?

What is the motivation behind axiomatically forcing the underpinning group of a ring to be abelian? Noncommutative rings are vastly more complex than commutative ones, so I am assuming that allowing ...
8
votes
3answers
520 views

Order of some quotient ring of Gaussian integers [duplicate]

I'm trying to get through a proof of Gauss' that certain primes can be written as the sum of two squares. An assumption is that the order of $\mathbb{Z}[i]/(a+bi)$ is $a^2+b^2$. I get that ...
8
votes
3answers
238 views

Ring Inside an Algebraic Field Extension

Let $E|F$ be an algebraic field extension and a ring $K$ such that $F\subseteq K\subseteq E$. It is true that $K$ is a field?
6
votes
3answers
901 views

Ring of order $p^2$ is commutative.

I would like to show that ring of order $p^2$ is commutative. Taking $G=(R, +)$ as group, we have two possible isomorphism classes $\mathbb Z /p^2\mathbb Z$ and $\mathbb Z/ p\mathbb Z \times \mathbb ...
3
votes
2answers
476 views

Proving a commutative ring can be embedded in any quotient ring.

Here's the exercise, as quoted from B.L. van der Waerden's Algebra, Show that any commutative ring $\mathfrak{R}$ (with or without a zero divisor) can be embedded in a ''quotient ring" consisting ...
2
votes
2answers
171 views

Can $R \times R$ be isomorphic to $R$ as rings?

I know from this question that $R \times R$ can be isomorphic to $R$, as $R$-modules. But can they ever be isomorphic as rings?
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votes
2answers
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A ring element with a left inverse but no right inverse?

Can I have a hint on how to construct a ring $A$ such that there are $a, b \in A$ for which $ab = 1$ but $ba \neq 1$, please? It seems that square matrices over a field are out of question because of ...
2
votes
1answer
142 views

for any ring $A$ the matrix ring $M_n(A)$ is simple if and only if $A$ is simple

Let integer $n\geq 1$. I have obtained that for any field $k$, the matrix ring $M_n(k)$ is simple, i.e., $M_n(k)$ contains no nonzero proper two sided ideals. Now I want to prove that: for any ring ...
26
votes
3answers
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A finite ring is a field if its units $\cup\ \{0\}$ comprise a field of characteristic $\ne 2$

Suppose $R$ is a finite ring (commutative ring with $1$) of characteristic $3$ and suppose that for every unit $u \in R\:,\ 1+u\ $ is also a unit or $0$. We need to show that $R$ is a field. Is this ...
6
votes
1answer
644 views

Irreducible Components of the Prime Spectrum of a Quotient Ring and Primary Decomposition

Recently I encountered a problem (the first exercise from chapter four of Atiyah & McDonald's Introduction to Commutative Algebra) stating that if $\mathfrak{a}$ is a decomposable ideal of $A$ (a ...
14
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2answers
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Is $A \times B$ the same as $A \oplus B$?

When $A, B$ are $K$-modules, then $A \times B$ is the same as $A \oplus B$. Let $A, B$ be two $K$-algebras, where $K$ is a field. Is $A \times B$ the same as $A \oplus B$? Thank you very much. ...
14
votes
3answers
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Complement of maximal multiplicative set is a prime ideal

Let $R$ be a commutative ring with identity. I've been trying to prove the following: If $S \subset R$ is a maximal multiplicative set, then $R \setminus S$ is a prime ideal of $R$. I have spent ...
6
votes
3answers
462 views

$1^n +2^n + \cdots +(p-1)^n \mod p =$?

Calculate for every positive integer $n$ and for every prime $p$ the expression $$1^n +2^n + \cdots +(p-1)^n \mod p$$ I need your help for this. I don't know what to do, but I'll show you what I ...
1
vote
2answers
476 views

Can we do long division in $\mathbb Z[\sqrt n]$, where $n$ is square free?

Can we do long division of elements $a + b\sqrt n$ and $c + d\sqrt n$ in $\mathbb Z[\sqrt n]$? I gave it a shot, but wasn't sure how to proceed... I'd like to use the gcd algorithm to show that two ...
14
votes
2answers
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Kaplansky's theorem of infinitely many right inverses in monoids?

There's a theorem of Kaplansky that states that if an element $u$ of a ring has more than one right inverse, then it in fact has infinitely many. I could prove this by assuming $v$ is a right inverse, ...
5
votes
4answers
543 views

Maximal ideals in $K[X_1,\dots,X_n]$

Let $K$ be a field, and $a_1,\dots,a_n \in K$. Prove that the ideal $$(X_1-a_1,\dots,X_n-a_n)$$ is maximal in $K[X_1,\dots,X_n]$. I tried proving that the only elements outside the ideal are the ...
2
votes
1answer
105 views

Transfer Between LCM, GCD for Rings?

I am starting a chapter on divisibility in commutative rings, and I was wondering if there was a way to translate theorems about gcd to lcm and vice versa. I know the concepts are considered "dual" in ...
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vote
6answers
745 views

$ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$ [closed]

How can we prove that $ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$?
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5answers
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Why “characteristic zero” and not “infinite characteristic”?

The characteristic of a ring (with unity, say) is the smallest positive number $n$ such that $$\underbrace{1 + 1 + \cdots + 1}_{n \text{ times}} = 0,$$ provided such an $n$ exists. Otherwise, we ...