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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1answer
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Exactly one ring homomorphism $F[X] \rightarrow S$

Let $F$ be a field, and $f \in F[X]/(f)$. Let $f$ have a zero point $\alpha$, that is, $f(\alpha)=0$. Let $F$ be a subring of $S$, and $\beta \in S$ with $f(\beta)=0$. Show that there is exactly one ...
4
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1answer
86 views

What is the kernel of $R[T] \to R[w]$, $T \mapsto w$, $w=u/v$, $u,v \in R$, where $R$ is an integrally closed domain?

I am posting the following question after posting a similar question: What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$? If $R$ is an integral domain, $w=u/v$, where $u,v ...
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3answers
40 views

Prime ideal in $\mathbb Z[\sqrt{10}]$

I am trying to solve this exercise: Prove that $\langle 2,\sqrt{10} \rangle$ is a prime ideal in $\mathbb Z[\sqrt{10}]$. I could do the following: I pick an element of the form $zw \in \langle ...
5
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1answer
86 views

What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$?

Consider $K[x^2,x^3] \subset K[x]$, where $x$ is an indeterminate over a (zero characteristic) field $K$. Clearly, $x$ vanishes the following polynomials $\in K[x^2,x^3][T]$: $f(T)=x^2T-x^3$, ...
7
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3answers
251 views

Looking for a Better Way to Think About Polynomial Rings

Given a commutative ring $R$, the polynomial ring in one variable $R[x]$ can be defined as the set of all the formal expressions $a_0+a_1x+\cdots+a_nx^n$ with 'obvious' rules of addition and ...
9
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2answers
90 views

For which $d \in \mathbb{Z}$ is $\mathbb{Z}[\sqrt{d}]$ a unique factorization domain?

Is there a general criterion which tells me whether $\mathbb{Z}[\sqrt{d}]$, $d \in \mathbb{Z}$ is a unique factorization domain? $\mathbb{Z}[\sqrt{-5}]$ is a frequent example for non-unique ...
3
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2answers
29 views

Action of the endomorphism ring of a module

Let $A$ be a ring with $1$ and $M$ a left $A$-module. In chapter 6 of Curtis and Reiner's 'Methods of Representation theory', $M$ is regarded as a right module over the endomorphism ring ...
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2answers
40 views

Flat modules and their relationship with short exact sequences

I recently came across the following result on a Wikipedia page: Suppose $0\to A\to B\to C\to 0$ is a short exact sequence where $B,\,C$ are flat modules; then $A$ is a flat module. I wanted to ...
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2answers
68 views

Is it true that a flat module is torsion free over an arbitrary ring? Does the reverse implication hold for finitely generated modules?

So when you work over a commutative ring, this result is quite well known. I am wondering if the same holds true for an arbitrary ring; that is, if $R$ is some (possibly noncommutative) ring, does the ...
0
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1answer
25 views

Boolean algebra-Boolean ring. Stone Theorem?

I am interested in knowing which theorem is responsible for the following statement: Every Boolean algebra can become a Boolean ring by taking the ring addition to be $A\oplus B = (A \land \lnot B) ...
0
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0answers
30 views

Why $m\phi=(m1)\phi=m(1\phi)=m1=m$?? [duplicate]

The is from a solution to prove that the only nonzero ring homomorphism from $$\mathbb{Q} \to \mathbb{R}$$ is the identity map that embeds $\mathbb{Q}$ in $\mathbb{R}$ as a subring and the solution ...
2
votes
1answer
36 views

Is $K[[x]]$ an Artinian/Noetherian $K[x]$-module?

Let $K$ be a field an consider $K[[x]]$ as a $K[x]$-module. Determine if it is Artinian/Noetherian. I used the following propositions: If M is an $R$-module and $N\subseteq M$ a submodule, then ...
6
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3answers
85 views

A commutative noetherian ring in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields

PROBLEM A commutative noetherian ring $R$ in which each ideal $I$ is principal and $I^2=I$ must be a finite product of fields. I am lost with the condition $I^2=I$ and the desired result "a ...
5
votes
1answer
43 views

Is $K[x_1,\ldots,x_{n+1}]$ separable over $K[x_1,\ldots,x_n]$?

Let $R \subseteq S$ be commutative rings. $S$ is separable over $R$ if $S$ is a projective $S \otimes_R S$-module (under $\mu: S \otimes_R S \to S$ defined by $\mu(s_1 \otimes s_2)=s_1s_2$). Let ...
2
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1answer
19 views

If $ R = L \oplus N $, then $ L = Re$ for idempotent $ e \in R $

I'm trying to prove the following statement. Suppose $ L \lhd R$ is a left ideal in $ R $ (a ring with unity). If there exists a left ideal $ N \lhd R $ such that $ R = L \oplus N $, then $ L = Re $ ...
3
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0answers
38 views

Left ideals in a subring of $M_2(R)$

Let $R$ be a 2-dimensional complete regular local ring $R$ over an algebraically closed field $k$, that is $R\cong k[[x,y]]$. Now look at the the following subring $A$ of $M_2(R)$: $A=\begin{pmatrix} ...
1
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1answer
33 views

A question on modules of rings

Before arising my question, let us see the following fact. Let $R$ be a ring with unity and $R'$ be a subring of $R$. If $M'$ is a simple $R'$-module, there exists a simple $R$-module $M$ such that ...
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1answer
23 views

Euclidean domains with multiplicative and super triangular norms

I want to prove that if the norm function $N$ of a Euclidean domain $R$ satisfies the conditions $N(ab)=N(a)N(b)$ $N(a+b) \le max\{N(a),N(b)\}$ then $R$ is a field or R is a ...
1
vote
1answer
33 views

show set is prime ideal

Let I = { (a,0): a E Z} A)show that I is a prime ideal of Z X Z B) by considering (ZXZ)/I , or otherwise , determine whether I is a maximal ideal of ZXZ. (0,0) is in I so I is non-empty let (a,0) ...
0
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0answers
21 views

free algebras over noncommutative rings

For a commutative ring $R$ and a set $X$, we can regard the polynomial algebra $R[X]$ as the free commutative $R$-algebra on $X$. For a unital associative ring $R$ which is not necessarily ...
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2answers
32 views

Prove that if the set of ideals is $\{\{0\}, R \}$, then $R$ is a field.

Let $R$ be an integral domain. Prove that if the set of ideals is $\{\{0\}, R \}$, then $R$ is a field. $\{0 \}$ and $R$ are the trival ideals of $R$. Let $I$ ba an ideal of $R$ and $a\in I$ ...
2
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1answer
34 views

Irreducibles and factorization in $\mathbb{Z}[\sqrt{5}i]$

Consider the ring $\mathbb{Z}[\sqrt{5}i]=\{m+n\sqrt{5}i:m,n\in\mathbb{Z}\}$. Show that $21$ has two distinct factorisations into irreducibles in $\mathbb{Z}[\sqrt{5}i]$, which is thus not a UFD. ...
1
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2answers
49 views

$\Bbb{Z}/10\Bbb{Z}$ isomorphic to $\Bbb{Z}[i]/\langle 1+3i\rangle$.

I need to prove that $\Bbb{Z}/10\Bbb{Z}$ is isomorphic to $\Bbb{Z}[i]/\langle 1+3i\rangle$. I know I can use the third isomorphism, but I would like to use the first one. I consider a homomorphism ...
3
votes
3answers
84 views

Prove $I$ is non-principal ideal of $\mathbb{Z}[x]$? [duplicate]

I'm new to algebra and got stuck with concept of ideals. The question is to prove that $$I = \left\{ {{a_0} + {a_1}x + \cdots + {a_n}{x^n} \mid {a_i} \in \mathbb{Z},{a_0} \in 2\mathbb{Z}} ...
2
votes
0answers
36 views

Inverse limit of the following system.

Let $\mathcal{O}_L = \mathbb{Z}[\sqrt{-5}]$ be the number ring of $L=\mathbb{Q}[\sqrt{-5}]$, and $\mathfrak{p} =(2, 1+\sqrt{-5})$ a prime (and maximal ideal) in $\mathcal{O}_L$. What is the inverse ...
2
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0answers
35 views

When is the set $A=\{a+s|a\in I , s\in S \}$ a prime ideal of R?

Let $R$ be commutative ring with identity, $I$ an ideal of $R$, and $S$ a subset of $R$. Under what conditions is the set $A=\{a+s\mid a\in I , s\in S \}$: 1- an ideal of $R$? 2- a prime ...
0
votes
1answer
15 views

Length of a ring? Lenth of a (right or left) ideal

I have seen the concept of length being applied to rings. What is exactly mean by that? What does length mean in a statement like "the composition length of RR is 2, but the composition length of RR ...
0
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0answers
54 views

In what conditions every ideal is an extension ideal?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). When $f$ is ...
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0answers
43 views

Extensions of Vector Spaces

I have just finished learning about field extensions and such. In particular I am interested in the minimum polynomial of $a$ in some field extension $K$ of $F$. Particularly I have learned about the ...
0
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1answer
24 views

Tensor product of commutative rings

I need help with this question: Suppose that A, B, C are commutative rings with unit. Is it true that $A\otimes_\mathbb{Z}(B\times C)$ is isomorphic as rings with $(A\otimes_\mathbb{Z} ...
5
votes
5answers
167 views

Minimal Polynomial of $\sqrt{2}+\sqrt{3}+\sqrt{5}$

To find the above minimal polynomial, let $$x=\sqrt{2}+\sqrt{3}+\sqrt{5}$$ $$x^2=10+2\sqrt{6}+2\sqrt{10}+2\sqrt{15}$$ Subtracting 10 and squaring gives ...
0
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1answer
28 views

Generating functions which are prime

Sorry for strangely worded title. The intended meaning is the generating functions which are not divisible by other generating functions, not functions for generating prime numbers. With this out of ...
0
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2answers
19 views

prime ideals, ring

If $ab \in 6\mathbb{Z}$, it does not follow that $a$ or $b$ is in $6\mathbb{Z}$. For example, $2 \cdot 3 = 6 \in 6\mathbb{Z}$, but $2$ nor $3$ is in $6\mathbb{Z}$. Can someone explain why? ...
15
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0answers
157 views

Can any commutative ring of characteristic $p\in\mathbb P$ be written as the form $R/(p)$ with $R$ being a ring of characteristic $0$?

Let $S$ be a commutative ring with identity with $\operatorname{char}S=p$, where $p$ is a prime number. I wonder if we can always find a ring $R$ such that $\operatorname{char}R=0$ and $R/(p)\cong ...
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0answers
30 views

Tensor products of ideals

Consider the class of complex algebras where the tensor products are over complex numbers. Given a complex algebra $A$ and a left ideal $L$ of $A$ generated by $n$ elements. Is $L^{\otimes n}$ ...
1
vote
1answer
44 views

If $p(x)\in F[x]$ is irreducible and $F$ has characteristic zero, then $p(x)$ has no repeated roots.

I want to prove that if $p(x)\in F[x]$, where $F$ is a field, is irreducible and $F$ has characteristic zero, then $p(x)$ has no repeated roots. I found this argument in a book, but I don't ...
2
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2answers
35 views

$(f(x))^p\neq f(x^p)$ on infinite field of characteristic $p$

It is easy to prove that if $f(x)\in\mathbb{Z}_p[x]$ then $(f(x))^p=f(x^p)$. Now, I want to show that this may be false if $\mathbb{Z}_p$ is replaced by an infinite field of characteristic $p$. The ...
5
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1answer
60 views

Separability of $A \subseteq C$ implies separability of $B \subseteq C$, where $A \subseteq B \subseteq C$

For commutative rings $R \subseteq S$, recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module. (via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$). My ...
2
votes
1answer
43 views

Ring theory associates

Can someone please give me an example of of this definition, as I am finding it hard to get my head around or even understand what an "associate" is. Let $R$ be a commutative ring with unity. ...
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1answer
23 views

If $I$ is a maximal ideal in $R$, $(I,x)$ is a maximal ideal in $R[x]$

Click Link to Original Text Let $R$ be a commutative ring with $1$, and $I$ is an ideal of $R$. Then, $(I) = I[x]$ is an ideal in $R[x]$. I was able to prove, via first isomorphism, that ...
0
votes
1answer
31 views

Trying to prove that for a ring $R$ with identity, $R^2 = R$

I'm trying to prove that if a ring $R$ has an identity, then $R^2 = R$. Honesty I don't even know where to start. Any help would be appreciated. Thank you very much, Eric
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0answers
31 views

What are some good books to study Non -Commutative Rings?

What are some good books to study Non -Commutative Rings? I want to study structure of semisimple rings and Wedderburn -Artin Theorem in particular . The book should provide motivations and have ...
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0answers
28 views

Lasker-Noether Theorem and Kummer-Dedekind

I would like to know about the relations between Ernst Kummer's invention of complex ideal numbers (and Dedekind's development of them into what is now called ideals) regarding the unique ...
3
votes
2answers
100 views

Show that $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a UFD. [duplicate]

I am trying to prove that the ring $\mathbb{C}[x,y]/(x^2+y^2-1)$ is a UFD. I have an hint, that suggests to find an isomorphism between $\mathbb{C}[x,y]/(x^2+y^2-1)$ and $\mathbb{C}[e^{it},e^{-it}]$, ...
1
vote
2answers
80 views

Show that, if $ a + bi$ is prime in $\mathbb{Z} [i]$, then $a - bi$ is prime in $\mathbb{Z}[i]$

Show that, if $ a + bi$ is prime in $\mathbb{Z} [i]$, then $a - bi $ is prime in $\mathbb{Z}[i]$ Since $\mathbb{Z}[i]$ is $ED$, then if $a+bi$ is irreducible then $a+bi$ is prime. But now how I ...
1
vote
1answer
22 views

Ring Homomorphisms from $\mathbb{Z} \to \mathbb{Z}/30\mathbb{Z}$

I am confusing myself here. Also, is it always understood that we are considering unital ring homomorphisms? $\phi(r)=\phi(\sum_1^r 1)= \sum_{i=1}^r \phi(1) = r \phi(1) = r(0+30 \mathbb{Z}) = r ...
3
votes
3answers
44 views

If $I$ and $J$ are ideals in $R$, and $I$ is a subset of $J$, is $I$ also an ideal in $J$?

Pretty much what the title suggests. If $I \subset J$ are both ideals in a commutative ring $R$, is it true that $I$ is an ideal in $J$? My reasoning for this is that clearly for all $a,b\in I$, $a ...
-1
votes
1answer
23 views

Tensor product of division rings is Noetherian

Let $k$ be a field and $D_{1},D_{2}$ division rings which are finite dimensional over $k$. Is it true that $D_{1} \otimes_{k} D_{2}$ is Noetherian? Can we say that yes since the tensor product is ...
0
votes
1answer
24 views

If I is an ideal of $R$ ($R$ a ring) prove that $M_{n}(R/I)$ is isomorphic to $M_{n}(R)/M_{n}(I)$

If $I$ is an ideal of $R$ ($R$ a ring) prove that $M_{n}(R/I)$ is isomorphic to $M_{n}(R)/M_{n}(I)$ I proved that $M_{n}(I)$ is an ideal of $M_{n}(R)$ but I don't know how to prove this. Thanks for ...
0
votes
2answers
23 views

Let $A,B\in \Bbb{K}^{n^2}, B\neq0$ prove that if $AB=0$ then $A$ is not invertible.

Let $A,B\in \Bbb{K}^{n^2}, B\neq0$ prove that if $AB=0$ then $A$ is not invertible. I'm having trouble proving this, I tried saying that $|AB|=|A||B|=0 \implies |A|=0 \text{ or } |B|=0$ but that got ...