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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5
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2answers
298 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
312 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
votes
1answer
266 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
vote
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
votes
2answers
364 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
votes
1answer
97 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?
1
vote
0answers
88 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
193 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
vote
1answer
181 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 ...
7
votes
3answers
319 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
618 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
votes
1answer
278 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
167 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
votes
0answers
53 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
348 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
125 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 ...
1
vote
1answer
102 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
vote
2answers
318 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
122 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
votes
0answers
49 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
103 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
votes
1answer
161 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 ...
1
vote
2answers
195 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 ...
2
votes
0answers
118 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
400 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
261 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 $r=r'.$ ...
2
votes
3answers
117 views

Interpreting $F[x,y]$ for $F$ a field.

First, is it appropriate to write $F[x,y] = (F[x])[y] $? In particular, if $F$ is a field, then we know $F[x]$ is a Euclidean domain. Are there necessary and sufficient conditions for when $F[x,y]$ is ...
2
votes
1answer
178 views

Is a surjective linear operator on $\mathbb{Z}^n$ always bijective?

For the simple case of $n=1$, this is pretty easy since if $\phi\in Hom(\mathbb{Z})$ is surjective then $\exists z \in \mathbb{Z}$ such that $\phi(z)=1$, since $\phi$ is linear we can factor ...
3
votes
1answer
274 views

What is the module $\operatorname{Hom}(M,N)$ where $R=\mathbb{C}[x]$ and $M=R/(x)$ and $N=R/(x-1)$.

Here is what I have so far: Consider $\mathbb{C}[x]/(x)$. This ring is all complex coefficient polynomials in $x$, and when quotient by the ideal $(x)$, we are subsituting $0$ for every $x$ in the ...
1
vote
1answer
120 views

Clarification on chinese remainder theorem for ring

http://www.math.uwaterloo.ca/~rdwillar/Courses/PM701/lecture_09_11.pdf I am not able to understand why $$(r_1 +\cdots + r_{i-1} + r_{i+1} +\cdots + r_k) - a_i \in A_i$$ in the last line of Theorem ...
2
votes
1answer
483 views

Definition of Ideals generated by a set

I'm struggling to understand the definition of ideals in ring homomorphisms generated by a set. If $R$ is commutative and has a $1$, then Ideal of $R$ generated by a subset $A$ of $R$: $$⟨ A ⟩ = ...
0
votes
3answers
558 views

find distinct elements of quotient ring of polynomials.

$2 + \langle x^3-x+5\rangle$ and $x^2 + \langle x^3-x+5\rangle$ are distinct elements of $\mathbb Z[x]/\langle x^3-x+5\rangle$ Is this true or false? I would be greateful if someone can explain the ...
4
votes
1answer
621 views

How to find Characteristic of quotient ring of polynomial.

How to find the characteristic if degree of f(x) is n? $$\mathrm{char}\Bigl(\mathbb{Z}[x]/\langle f(x)\rangle\Bigr)$$
5
votes
1answer
176 views

How to view set of equivalence classes of extensions of M by N as an A-module

I know that for a commutative ring $A$ and $A$-modules $M$ and $N$, the set $E_A(M, N)$ of extensions of $M$ by $N$ can be equipped with the Baer sum which gives it an additive group structure. ...
4
votes
2answers
419 views

Does a homomorphism from a unital ring to an integral domain force a multiplicative identity?

This is a question in Herstein's Topics in Algebra ("unit element" refers to multiplicative identity): If $R$ is a ring with unit element $1$, and $\phi$ is a homomorphism of $R$ into an integral ...
7
votes
2answers
148 views

Countable rings

Suppose we are given a countable unital ring $R$ with uncountably many distinct right ideals. Does it follow from this that $R$ has uncountably many maximal right ideals?
1
vote
1answer
218 views

Help determine the group of units of the ring $\mathbb{Z}[(1+\sqrt{m})/2]$

let $m$ be an integer with $m\equiv 1 \pmod4$ and $m<-3$. $U\left(\mathbb{Z}+\mathbb{Z}(\frac{1+\sqrt{m}}{2})\right)=\{\pm1\}$ How can I prove that?
3
votes
0answers
214 views

Noetherian rings, why commutativity?

I am looking for an answer to why one has to assume commutativity of a ring $R$ in proving some results about Noetherian rings. For example, Let $R$ be a commutative ring; look at the proof(s) of the ...
0
votes
2answers
120 views

Two ring questions

In the ring $R=\mathbb Z[X],$ is $(X)+(X^2)=(X)$? It is known that if $R$ is a UFD, then $R[X]$ is a UFD. Is the converse true?
2
votes
1answer
242 views

Rings such that $A[x]$ is a principal ideal domain

Let $A$ be a commutative ring. Then the following assertions are equivalent. $A$ is a field; $A[x]$ is a Euclidean domain; $A[x]$ is a principal ideal domain; $A[x]$ is a unique factorization ...
8
votes
1answer
382 views

A graded ring $R$ is graded-local iff $R_0$ is a local ring?

Update: I've copied this question over to mathoverflow.net: http://mathoverflow.net/questions/100755/a-graded-ring-r-is-graded-local-iff-r-0-is-a-local-ring to see if I get any answers there. Let ...
0
votes
1answer
92 views

Quotient map is not a retraction

Please help me to solve the following problem. Let $L$ be a left ideal in a unital ring $R$, such that $L^2\neq L$. Prove that quotient map $\pi:R\to R/L$ have no right inverse homomorphism of ...
2
votes
1answer
192 views

How many elements does $R$ have?

Let $f: R\to S$ be a homomorphism of rings (with $R$ commutative) such that kernel of $f$ has $4$ elements and image of $f$ has $16$ elements. How many elements does $R$ have? Would you simply use ...
1
vote
0answers
108 views

Ring of invariants for cyclic matrix group acting on polynomial ring?

If $G \leq GL_2(\mathbb{C})$ is generated by matrix $\begin{pmatrix} 1 & 2\\0 & -1 \end{pmatrix}$, acting on the polynomial ring $\mathbb{C}[X,Y]$, then how can we find the ring of invariants ...
1
vote
1answer
460 views

Describe the ring and the cosets.

Describe the ring $R = \mathbb Z_4[x]/((x^2+1)\mathbb Z_4[x])$ by listing all the cosets (for example by using coset representatives) describing the relations that hold between the elements in ...
10
votes
2answers
463 views

Prove that an infinite ring with finite quotient rings is an integral domain

How can we show that if $R$ is an infinite commutative ring and $R/I$ is finite for every nontrivial $I \unlhd R$, then $R$ is an integral domain? I tried proceeding by contradiction: assume ...
1
vote
1answer
158 views

comaximality of ideals in a commutative ring with unit

Suppose we have a commutative ring $R$ with unit. I'm curious about what condition(s) on $R$ would be sufficient (without Axiom of Choice) to give a converse to the following familiar result: (#) If ...
3
votes
2answers
313 views

Isomorphism $k[x,y]/(y-x^2)$ onto $k[x]$

Perhaps this is very easy but I would like to get a proof of isomorphism $k[x,y]/(y-x^2) \cong k[x]$.
2
votes
3answers
332 views

Showing that $\sqrt5$ is not in $\mathbb{Q}(\sqrt7)$

How can I prove that $\sqrt5$ is not in $\mathbb{Q}(\sqrt7)$ ? I can only think of trying to write $\sqrt5 = a+b\sqrt7$ (where $a,b$ are in $\mathbb{Q}$), but I can't think of a good reason that ...