This tag is for questions about rings, which are an algebraic structure studied in abstract algebra and algebraic number theory.

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165 views

Is a ring closed under both operations?

A ring is a set R, together with two binary operations $+, \cdot : R\times R \to R$ that satisfy $(R,+)$ is an abelian group Associativity Distributivity Multiplicative identity so $\exists 1_R ...
3
votes
1answer
482 views

Zero divisor, nilpotent elements of quotient ring $\mathbb Z_2[x] / \langle x^8-1\rangle$

Consider the ring $R = \displaystyle\frac {\mathbb Z_2[x]}{\langle x^8-1\rangle}$. i) Is $R$ a finite ring? ii) Does $R$ have a zero divisor? iii) Does $R$ have nilpotent elements? ...
6
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3answers
3k 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 ...
1
vote
1answer
177 views

If $R$ is a finite dimensional algebra over a field then $R$ is simple as a ring if and only if it has a faithful simple left $R$-module

How can I prove that if $R$ is a finite dimensional algebra over a field then $R$ is simple as a ring if and only if it has a faithful simple left $R$-module?
4
votes
2answers
266 views

Showing polynomials in $k[x_1, \ldots , x_n]$ are irreducible

It is often the case when I wish to show a particular polynomial in $k[x_1, \ldots ,x_n]$ is irreducible. Assuming that the polynomial is sufficiently friendly (i.e. one I would encounter as part of a ...
4
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3answers
267 views

Rings with isomorphic proper subrings

Rings will be unital here but I don't require that subrings share the identity elements with superrings. I just accidentally came up with an example of a ring $R$ with a proper subring $S$ such that ...
3
votes
2answers
162 views

Surjective graded homomorphism of rings also an isomorphism?

Suppose we are given two graded (commutative) rings $A$ and $B$ and a graded homomorphism $\psi:A\rightarrow{B}$ between them. Suppose moreover that $\psi$ is surjective in each degree i.e. that ...
5
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3answers
140 views

Is this action of $\mathbb F[[x]]$ on $\bigoplus_{i=0}^{\infty}\mathbb F$ natural?

The title of my question has a field $\mathbb F$ in it, but to make sure I'm not losing anything, I would like to introduce my question in full generality. But still, I will be happy with an answer in ...
2
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1answer
101 views

Associative division subalgebras of split Cayley-Dickson algebra

Let's consider the split Cayley-Dickson algebra $C$ over an arbitrary field $F$ (It is well known that all split composition algebras having the same dimension over base field are isomorphic, e.g., ...
6
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0answers
115 views

On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.

I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
2
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2answers
183 views

What if $\operatorname{char}\mathbb{K}$ is not $0$ or if $\mathbb{K}$ is not algebraically closed? (Nullstellensatz)

Given a field $\mathbb{K}$ which is algebraically closed and of characteristic 0, we can say exactly what the maximal ideals of $\mathbb{K}[x_1,\dots,x_n]$ are and they correspond to points in ...
2
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0answers
61 views

Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
2
votes
1answer
90 views

Any endomorphism of $K[x]$ which is the identity on $K$ is $E_g$ for some $g$ in $K[x]$

Prove: Any endomorphism of $K[x]$ which is the identity on $K$ is $E_g$ for some $g$ in $K[x]$. Note: In my book it defines $E_g\colon K[x] \to K[x]$ by sending $x$ to $g$. This seems like it ...
3
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3answers
354 views

The intersection of distinct maximal ideals is not prime.

Let $P,Q$ be distinct maximal ideals of a ring $R$. Prove that $P\cap Q$ is not prime. I am not sure how to prove this. The only facts that I can think of applying are the definitions, $R/M$ is ...
3
votes
3answers
984 views

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 $yx=1$?
0
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2answers
229 views

Sided inverses in a non-commutative ring

I've asked myself the following question : does there exist a non-commutative ring $R$ with unity $1$ and elements $x,y,z \in R$ such that $xyz = 1$ but $y$ has no left nor right inverses? (Perhaps I ...
3
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3answers
108 views

Are the $\mathbb{Z}/p^k\mathbb{Z}$ simple modules all isomorphic to one another?

After looking back over some finite field theory, I've been thinking about the ring $\mathbb{Z}/p^k\mathbb{Z}$ for some prime $p$. I'm just curious, are the $\mathbb{Z}/p^k\mathbb{Z}$ simple modules ...
5
votes
1answer
194 views

Is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$?

If $p\in\mathbb{N}$ is a prime, is $x^n+px+p^2$ irreducible in $\mathbb{Z}[x]$? I've proved that any non-unit factor in $\mathbb{Z}[x]$ must have degree at least 2. Eisenstein's criterion doesn't ...
0
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1answer
137 views

What is the relation between prime ring and indecomposable ring?

Prime ring means $0$ is prime in the ring, so $R$ is a prime ring iff for any two elements $a$ and $b$ of $R$, $arb = 0$ for all $r\in R$ implies that either $a = 0$ or $b = 0$. Indecomposable ring ...
13
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2answers
917 views

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 ...
1
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2answers
234 views

Followup to “Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$”

In this post: Examples of rings with ideal lattice isomorphic to $M_3$, $N_5$ a nice example was given of a non-distributive ring. The lattice of ideals turned out to be the Diamond lattice $M_3$ with ...
1
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0answers
105 views

Unit group of quotient of noncommutative polynomial ring

In this recent post the original question led people to look for rigid, noncommutative rings. (Rigid means that the only endomorphisms are zero and the identity). Several (somewhat complicated) ...
0
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1answer
119 views

Prove that $[R:I] =\{r \in R\mid xr \in I\text{ for every }x \in R\}$ is an ideal of $R$ that contains $I$

If $I$ is an ideal in a ring $R$ let $[R:I] =\{r \in R\mid xr \in I\text{ for every }x \in R\}$. How can I show that $[R:I]$ is an ideal of $R$ which contains $I$.
5
votes
2answers
287 views

certain examples of fields of fractions

Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of ...
5
votes
1answer
283 views

Ideals in non-associative rings and the identity $(xy)z=y(zx)$.

I have come across this paper. The authors prove that magmas satisfying the identity $$(xy)z=y(zx)\tag1$$ are nearly both associative and commutative. To be precise, they show that in such magmas, ...
14
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1answer
247 views

Ideal in an Artinian Ring $I=aR=Rb$, prove $I=Ra=bR$

Let $R$ be an Artinian Ring and suppose there exists $a,b\in R$ s.t. $I=aR=Rb$, then prove $I=bR=Ra$. (You may assume that a right Artinian Ring is Right Noetherian). I've managed to get $Ra$, ...
1
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0answers
45 views

$GL_2$-Invariants of $\mathbb{C}[X,Y]$

One of the problems in some work I'm doing tells me to consider $GL_2$ acting on $\mathbb{C}[X,Y]$, induced by the natural representation of $GL_2$ on $\mathbb{C}^2$. I just wanted to check 2 things: ...
2
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2answers
343 views

How to show that if two integral domains are isomorphic, then their corresponding field of quotients are isomorphic?

If two integral domains $D$ and $D'$ are isomorphic show that their corresponding field of quotients (fractions) $Q(D)$ and $Q(D')$ are isomorphic.
2
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1answer
92 views

Is $\bigwedge(V)$ self-injective?

For a vector space $V$, is the grassman algebra $\bigwedge(V)$ always an injective module over itself? Is there a proof, or even just a brief explanation?
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0answers
80 views

Question on Bergman's Diamond Lemma.

Last week I was recommended Bergman's Diamond Lemma in these comments. I read through the paper, and was working on an exercise in it. I want to know if the reduction systems $\{(x^2y^2,yx)\}$ and ...
6
votes
2answers
183 views

An $R$ module and $S$ module that cannot be an $R$-$S$ bimodule

In connection with this question: Modules and tensor products Question: For two commutative rings $R$ and $S$ (with unity), is there an abelian group $M$ which has $R$ module and $S$ module ...
1
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1answer
171 views

Example request: simple, radical ring

I'm looking for an example of a ring $R$ (necessarily nonunital) which is simple (in the sense that $R \cdot R \neq 0$ and $R$ has no proper, nonzero 2-sided ideals) and also radical (in the sense ...
6
votes
3answers
303 views

Module M/IM of finite length $\implies$ Ring A/I of finite length

This question is due to a proof in an algebra book (on the topic of dimension theory) which I don't fully understand (specifically, the proof of Thm 6.9b) in Kommutative Algebra by Ischebeck). It may ...
10
votes
1answer
590 views

Prove that $\mathbb{C}[x,y] \ncong \mathbb{C}[x]\oplus\mathbb{C}[y]$

Prove that $\mathbb{C}[x,y] \ncong \mathbb{C}[x]\oplus\mathbb{C}[y]$ $\mathbb{C}[x,y]$ is the polynomial ring of two variables over $\mathbb{C}$. I guess that we can consider images of $xy$ and ...
11
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1answer
258 views

What are the rings in which left and right zero divisors coincide called?

A unital ring $R$ is reversible iff $ab=0\implies ba=0.$ This condition implies the following one. If $a\in R$ is a left-zero divisor, then $a$ is also a right-zero divisor. And the other way ...
3
votes
3answers
150 views

Semisimplicity of a polynomial ring

Given a ring $R$ (with 1 and not necessarily commutative) when is the polynomial ring $R[x]$ semisimple? For example if R is a Noetherian integral domain then R[x] is not semisimple. Indeed, $R[x]$ ...
0
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0answers
49 views

Does $\{(yx,x^2y)\}$ admit a normal form?

This question is a follow up to the question I asked here a few days ago. Suppose the reduction rule given is $(yx,x^2y)$, that is, the rule is switched around as was suggested in the comments. Does ...
42
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 ...
7
votes
2answers
318 views

Ideal as a projective module

I am sorry, this may not be a good question here. I am looking a good reference about when the ideal $I$ of a given commutative ring $R$ (local or may not be local) with identity is a projective ...
1
vote
1answer
122 views

Ring Theory Least Common Multiples

If $a$ and $b$ are elements in an integral domain with unity 1$\neq$0. Show that $a$ and $b$ have a least common multiple if $a$ and $b$ have a highest common factor. More generally there is a ...
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1answer
100 views

ring theory - basic question, it seems

We're in an integral domain with unity 1 $\neq$ 0. Suppose that the highest common factor between x,y is 1 and the highest common factor for x,z is 1. Show that $x \mid yz$ implies that $x$ is a ...
1
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2answers
305 views

Irreducible Polynomial in Field of 2 Elements?

How do I show that $ f(t) = t^2 + t +1 $ is irreducible in $K[t]$, where $K = \{0,1\}$? I know how to tackle this over $\mathbb{Z}$ or $\mathbb{Q}$ using Guass or Eisenstein say...but I'm a little ...
2
votes
1answer
118 views

Is the functor $R \mapsto \mathbb{M}_n(R)$ a right adjoint?

Given a positive integer $n$, there is a functor $F: \mathsf{Ring} \rightarrow \mathsf{Ring}$ such that $F(R) = \mathbb{M}_n(R)$ on objects and the action of $F$ on morphisms are given entrywise. Is ...
3
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0answers
48 views

Does the singleton reduction system $\{(x^2y,yx)\}$ lead to a normal form?

Suppose you have a singleton reduction system $\{(x^2y,yx)\}$. Does such a system lead to a normal form on the corresponding $k$-algebra $k\langle x,y\rangle$, where $k$ is a commutative, ...
0
votes
1answer
93 views

Why is the identity a finite sum of elements in this left ideal decomposition?

Suppose you have a semisimple ring $R$, and want to decompose it into a sum of simple left ideals. Let $\{L_i\}$ be a family of simple left ideals, such that no two are isomorphic, and any simple left ...
3
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1answer
147 views

Two questions about integral “splitting ring” extensions

We have a ring $R$, commutative, and $f_1,\dots,f_n$ polynomials in $R[x]$ monic, with $\deg f_i\ge 1$. It is straightforward to show that there is a ring extension $R\subset S$ such that $S$ contains ...
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2answers
193 views

What is $\langle x + y\rangle + \langle x - y \rangle$?

Let $R$ be a ring. Let $I = \langle x + y \rangle$, $J = \langle x - y \rangle$ be ideals of $R[x,y]$. What's $I + J$ in this case? By definition $I + J = \{ i + j \mid i \in I, j \in J \}$. My first ...
1
vote
0answers
111 views

Dimension of a module vs codimension of its annihilator

For some reason I am just stuck on this question, although my intuition insists it's easy - I'd appreciate anyone telling me any obvious fact I'm overlooking here. Suppose we have an ...
3
votes
2answers
387 views

Prove that $\mathbb{Z} [x] $ is not isomorphic to $\mathbb{Q}[x,y]$.

I know they are both UFDS, but not Euclidean domains nor PIDS. The argument that I have seen showing that they are not isomorphic goes along the lines of saying the number of invertible elements in ...
0
votes
2answers
244 views

The structure of the kernel of a ring homomorphism

I want to show that given a ring homomorphism $\phi:R\to S,$ if $\phi$ is injective, then $\ker(\phi)=\{0\}.$ Given $\phi(r)=\phi(r'),$ by the definition of an injection, we have $r=r'.$ ...