# Tagged Questions

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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### 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 ...
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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 ...
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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. ...
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### 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: ...
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### 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 ...
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### 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 ...
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### $\mathbb C[X]/(X^2)$ 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 $\mathbb R[Y]/((Y^2+1)^2)$.
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### Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ 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 able ...
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### Does every ring of integers sit inside a ring of integers that has a power basis?

Given a finite extension of the rationals, K, we know that $K=\mathbb{Q}[\alpha]$ by the primitive element theorem, so every $x \in K$ has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n$$ ...
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### Is Tolkien's Middle Earth flat?

In the first introductory chapter of his book Gravitation and cosmology: principles and applications of the general theory of relativity Steven Weinberg discusses the origin of non-euclidean ...
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### 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 ...
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### 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 $Y'$ ...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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### 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). E.g....
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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 ...
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### 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 $r$...
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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.
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### How to prove that a complex number is not a root of unity?

$\frac35+i\frac45$ is not a root of unity though its absolute value is $1$. Suppose I don't have a calculator to calculate out its argument then how do I prove it? Is there any approach from ...
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### Does integration by parts with “deja vu” have a name?

In some integration by parts problems, such as evaluating the integral of $e^x \cos x$ or $\sec^ 3 x$, one performs integration by parts (possibly more than once, and possibly together with algebraic ...
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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 $a\in ... 4answers 898 views ### What are the zero divisors of$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)$. I'm ... 2answers 3k views ### necessary and sufficient condition for trivial kernel of a matrix over a commutative ring In answering Do these matrix rings have non-zero elements that are neither units nor zero divisors? I was surprised how hard it was to find anything on the Web about the generalization of the ... 2answers 753 views ### What is the coproduct of fields, when it exists? This is a slightly more advanced version of another question here. Let$\textbf{CRing}$be the category of commutative rings with unit. Let$\textbf{Dom}$be the category of integral domains – by ... 3answers 16k views ### what is difference between a ring and a field The ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition. A field can be thought of as two groups with extra ... 1answer 206 views ### Does$A^2 \cong B^2$imply$A \cong B$for rings? If$A$and$B$are two unital rings such that$A \times A \cong B \times B$, as rings, does it follows that$A$and$B$are isomorphic (as rings)? I believe that the answer is no, but I can't come ... 1answer 697 views ### When is a group ring an integral domain If$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$then$(e-g)(e+g\...
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 ...
Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.