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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172
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A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
80
votes
4answers
2k views

Does $R[x] \cong S[x]$ imply $R \cong S$?

This is a very simple question but I believe it's nontrivial. I would like to know if the following is true: If $R$ and $S$ are rings and $R[x]$ and $S[x]$ are isomorphic as rings, then $R$ and $S$ ...
56
votes
5answers
3k views

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 ...
49
votes
12answers
5k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
46
votes
4answers
2k views

Algebra: Best mental images

I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
45
votes
6answers
1k views

Is there a non-commutative ring with a trivial automorphism group?

This question is related to this one. In that question, it is stated that nilpotent elements of a non-commutative ring with no non-trivial ring automorphisms form an ideal. Ted asks in the comment for ...
40
votes
2answers
3k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
35
votes
1answer
494 views

A ring isomorphic to its finite polynomial rings but not to its infinite one.

I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious: ...
34
votes
2answers
2k views

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 ...
33
votes
6answers
4k views

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 ...
32
votes
3answers
3k views

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) = ...
32
votes
1answer
2k views

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. ...
32
votes
4answers
921 views

$\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$

This question led me to the following: Prove that $\mathbb C[X]/(X^2)$ is isomorphic to $\mathbb R[Y]/((Y^2+1)^2)$.
29
votes
5answers
7k views

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 ...
29
votes
4answers
1k views

Fermat's Last Theorem and Kummer's Objection

In 1847 Lamé had announced that he had proven Fermat's Last Theorem. This "proof" was based on the unique factorization in $\mathbb{Z}[e^{2\pi i/p}]$. However, Kummer, proved that when $p=23$ we do ...
29
votes
3answers
1k views

Ideals of $\mathbb{Z}[X]$

Is it possible to classify all ideals of $\mathbb{Z}[X]$? By this I mean a preferably short enumerable list which contains every ideal exactly once, preferably specified by generators. The prime ...
26
votes
6answers
3k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
26
votes
2answers
5k views

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 ...
26
votes
3answers
1k views

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 ...
25
votes
9answers
2k views

Are there rings whose multiplicative identity is not the number 1 or number 1-based?

Reading the basic definition of rings, I wondered if there are samples of rings whose multiplicative identity is not the number 1 or number 1-based (for instance the identity matrix is 1-based). ...
23
votes
3answers
2k views

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 ...
23
votes
3answers
557 views

A Laskerian non-Noetherian ring

A Laskerian ring is a ring in which every ideal has a primary decomposition. The Lasker-Noether theorem states that every commutative Noetherian ring is a Laskerian ring (as an easy consequence of the ...
22
votes
5answers
1k views

Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...
22
votes
7answers
5k views

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 ...
22
votes
4answers
712 views

Why do we study prime ideals?

I hope this isn't an inappropriate question here! I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? ...
22
votes
1answer
638 views

$Ra=Rb$ if and only if $aR=bR$

On which classes of (non commutative) rings we have the following property: $aR=bR$ if and only if $Ra=Rb$ ? While I googling around I found the notion of "Duo Ring" in which $aR=Ra$ for every $a\in ...
21
votes
7answers
1k views

Does every unital ring contain all the integers?

Let us suppose there is a ring $R$ with the multiplicative identity $1$. We know that $1+r\in R$, where $r$ is any element of the ring $R$. Does this mean $1+1$ is also part of the ring, or does ...
21
votes
1answer
771 views

A ring with few invertible elements

Let $A$ be a ring with $0 \neq 1 $, which has $2^n-1$ invertible elements and less non-invertible elements. Prove that $A$ is a field.
21
votes
4answers
724 views

What are the zero divisors of $C[0,1]$?

Suppose you have a ring $(C[0,1],+,\cdot,0,1)$ of continuous real valued functions on $[0,1]$, with addition defined as $(f+g)(x)=f(x)+g(x)$ and multiplication defined as $(fg)(x)=f(x)g(x)$. I'm ...
21
votes
2answers
1k views

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
20
votes
2answers
667 views

What is the coproduct of fields, when it exists?

This is a slightly more advanced version of another question here. Let $\textbf{CRing}$ be the category of commutative rings with unit. Let $\textbf{Dom}$ be the category of integral domains – by ...
20
votes
0answers
409 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
19
votes
12answers
3k views

Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.

I am trying to prove that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic. I was thinking of arguing the following: Suppose there exists an isomorphism $\varphi: ...
19
votes
3answers
747 views

What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?

Let $R$ be a commutative ring with a unit. $\newcommand{\spec}{\operatorname{Spec}}\spec(R)$ denotes the set of all prime ideals in $R$, and it can be topologized using the Zariski topology. Last ...
19
votes
2answers
2k views

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
2answers
3k views

What do prime ideals in $k[x,y]$ look like?

Suppose that $k$ is an algebraically closed field. Then what do the prime ideals in the polynomial ring $k[x,y]$ look like? As far as I know, the maximal ideals of $k[x,y]$ are of the form ...
18
votes
1answer
250 views

How 'commutative' can a non-commutative ring be?

Let $R$ be a finite non-commutative ring. Let $P(R)$ be the probability that two elements chosen uniformly at random commute with each other. Consider the value $$S=\sup_RP(R)$$ where the supremum ...
18
votes
1answer
528 views

When is a group ring an integral domain

If $R$ is an integral domain (I am having $\mathbb{Z}$ or a field in mind) and $G$ a (not necessarily finite) group then we can form the group ring $R(G)$. Note that if $g^{n+1} = e$ then ...
17
votes
5answers
1k views

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 ...
17
votes
5answers
3k views

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.
17
votes
4answers
1k views

Does $ab=1$ imply $ba=1$ in a ring?

Let $R$ be a ring with unity $1$ and let $a,b\in R$ be not zero divisors. Is there any counterexample for: $$ab=1\quad\Rightarrow \quad ba=1$$
17
votes
3answers
834 views

Ring such that $x^4=x$ for all $x$ is commutative

Let $R$ be a ring such that $x^4=x$ for every $x\in R$. Is this ring commutative?
17
votes
3answers
937 views

Motivation for Eisenstein Criterion

I have been thinking about this for quite sometime. Eisentein Criterion for Irreducibility: Let $f$ be a primitive polynomial over a commutative unique factorization domain $R$, say $$f(x)=a_0 + ...
17
votes
4answers
636 views

Pathologies in module theory

Linear algebra is a very well-behaved part of mathematics. Soon after you have mastered the basics you got a good feeling for what kind of statements should be true -- even if you are not familiar ...
17
votes
2answers
6k views

Khan academy for abstract algebra

I am looking for instructional videos for abstract algebra, specifically topics including group theory, ring theory, isomorphic and homomorphic structures, and properties of groups and rings, and ...
17
votes
2answers
752 views

Proof that a certain subset of the reals is not a ring

Let $A = \{x \sin x : x \in \mathbb{Z}\} \subset \mathbb{R}$. Is $A$ a ring under the usual addition and multiplication operations of $\mathbb{R}$? It looks like it's not, but I can't find something ...
16
votes
8answers
7k views

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.
16
votes
7answers
791 views

Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
16
votes
4answers
951 views

Smallest non-commutative ring with unity

Find the smallest non-commutative ring with unity. (By smallest it means it has the least cardinal.) I tried rings of size 4 and I found no such ring.
16
votes
6answers
3k views

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 ...