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
87 views

If $R$ contains two ideals $B$ and $C$ with $B+C=R$ and $B \cap C=0$, then $B$ and $C$ are rings and $R\cong B\times C$

If $R$ contains two ideals $B$ and $C$ with $B+C=R$ and $B \cap C=0$, then $B$ and $C$ are rings and $R\cong B\times C$ I tried to prove it using definition of subrings, that is, B and C is ...
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2answers
132 views

Calculation of radical ideal in $\mathbb Z_{36} $

Let $R$ be the ring $\mathbb Z_{36}$. How can I calculate $ \sqrt{\langle 0\rangle} , \sqrt{\langle 9\rangle} $?
5
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2answers
237 views

Given a prime $p\in\mathbb{N}$, is $A=\frac{x^{p^{2}}-1}{x^{p}-1}$ irreducible in $\mathbb{Q}[x]$?

If $p \in \mathbb{N}$ is a prime, is $\displaystyle A=\frac{x^{p^{2}}-1}{x^{p}-1}$ irreducible in $\mathbb{Q}[x]$? I don't think it is. If somebody sees a contradiction, I would be glad to see it. ...
0
votes
1answer
214 views

Proving something is Noetherian

Let say you have a ring $R={\left(\begin{array}{cccc} \mathbb{C}[x,y] & y^2\mathbb{C}[x,y]\\ xy\mathbb{C}[x,y] & \mathbb{C}[x,y] \end{array} \right)}$ It it enough to prove that it's a ...
6
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2answers
206 views

Show that there is an infinite amount of reducible polynomials of the form $x^n+x+1$ in $\mathbf{F}_2$

Here is a question from an old exam: Show that there are infinite $n\in \mathbf{N}, A= x^{n}+x+1 $ which are reducible over $\mathbf{F}_{2}[x]$. Using André Nicolas' and Qiaochu Yuan's hint: ...
0
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1answer
71 views

unit u and primes in $\mathbb{Z}[i]$ with criterias

There is a unit u and primes $\pi_{j}=a_{j}+b_{j}i$ in $\mathbb{Z}[i]$ with $a_{j}>0, b_{j}>0$ and $7+i = u\pi_{1}\dots\pi_{k}$ $\mathbb{Z}[i]$ has four units: $i,-i,1,-1$. The product ...
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3answers
94 views

Given a UFD and two polynomials not equal to zero, product of gcd is a unit

Here is a question from an old exam: 1. a) Let $R$ be a UFD and let $A=a_{0}x^{m}+...+a_{m}\ne 0 , B=b_{0}x^{n}+...+b_{n} \ne 0$ in $R[x]$ with $\gcd(a_{0},...,a_{m})\in R^{*}$, ...
2
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1answer
104 views

Working out what generates $\mathbb{C}[x,y]$?

Is $\mathbb{C}[x,y]$ finitely generated $\mathbb{C}$-algebra? Also is it 2-generated? As I can't see the reason why this is true, yet we are using reasoning like this in a course in non commutative ...
2
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4answers
832 views

$a^m=b^m$ and $a^n=b^n$ imply $a=b$

Let $D$ be an integral domain and let $a^m=b^m$ and $a^n=b^n$ where $m$ and $n$ are relatively prime integers, $a,b \in D$. How do I show $a=b$?
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1answer
279 views

If $x^4 =x$ in a ring, then the ring is commutative. [duplicate]

Possible Duplicate: Ring such that $x^4=x$ for all $x$ Let $R$ be a ring such that $a^4=a$ $ ,\forall a \in R$. How do I show that $R$ is commutative?
0
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1answer
150 views

Radicals in a ring

I have stumbled across the following exercise on radicals of ideals of rings. I shall show that: $\operatorname{rad}(x+y^2,x^2+2xy^2)$ is a maximal ideal of $\mathbb{C}[x,y]$, but $(x+y^2,x^2+2xy^2)$ ...
0
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1answer
113 views

Visualisation of polynomials in a quotient ring

Let $I=[x,y]$ be the prime ideal generated by the polynomials $x,y$ with real coefficients and let $R_I$ be the localization of the ring $R=\mathbb{R}[x,y]$ in $I$. Can someone help my to visualize/to ...
4
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2answers
4k views

GCD of gaussian integers

Let $\mathbb Z [i] =\{a+bi: a,b \in \mathbb Z\}$. What is the gcd of $11+7i$ and $18-i$ in $\mathbb Z [i]$?
2
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2answers
150 views

an invertible element $i$ in $\mathbb Z_n$ must be coprime to $n$

Let $n$ be an integer and $i\in \{1,\cdots,n-1\}$. I want to show that $i$ is invertible in $\mathbb Z_n$ if and only if $i$ is coprime to $n$. One way is easy. suppose $i$ is coprime to $n$ then ...
2
votes
1answer
124 views

A property of different in Dedekind domains

Let $A \subseteq B$ be a finite extension of Dedekind domains such that the extension $K \subseteq L$ of their quotient fields is separable. Let $\mathfrak{p}$ be a maximal ideal of $A$ and let ...
5
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2answers
659 views

How to show that $R/I$ is Artinian when R is PID

I'm working through some of Hungerfords "Algebra", and having trouble with Excercise VIII 1.2.: Show that if $I$ is a non-zero ideal in a principal ideal domain (PID) $R$, then the ring $R/I$ is ...
1
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1answer
236 views

Automorphisms of $F[x]$

What are the automorphisms $\sigma$ of $F[x]$ with the property that $ \sigma(f) =f $ for every $f \in F$? $F$ is a field here.
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2answers
263 views

Additive inverse of a nilpotent element is nilpotent

An element of a ring $R$ is nilpotent if $a^n=0$ for some $n \ge 1$. How do I show that additive inverse of $a$ , $-a$ is also nilpotent? The ring is commutative but may not have a unit element.
3
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2answers
235 views

A doubt about Gauss's lemma

Gauss's lemma says If the primitive polynomial $f(x)$ can be factored as product of two polynomials having rational coefficients, it can be factored as the product of two polynomials having integer ...
-1
votes
1answer
291 views

Existence of GCD in an integral domain

What is the necessary and sufficient condition for an integral domain to have gcd for every pair of elements and why?
1
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1answer
318 views

How to prove that the ring of p-adic integers is an Euclidean domain?

Let $ p \in \mathbb{Z}$ be a prime, and define $$R:=\lbrace a=(a_1,a_2,a_3,\ldots) | a_k \in(\mathbb{Z}/p^k\mathbb{Z})\text{ and }a_{k+1}\equiv a_k \pmod {p^k}\text{ for all }k \in \mathbb{N} ...
2
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1answer
226 views

Proving that the ring is not an euclidean domain

Consider the ring $R= \mathbb Z [(1+\sqrt-19)/2]$. How do I prove it is not an euclidean domain?
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0answers
84 views

What is a “staircase” of an ideal in a polynomial ring?

I've come across the term "staircase" applied to an ideal in a polynomial ring over a field. Can someone explain this? If a bit more formalism is required: Let $k$ be a field, let $R = k[X_1, \ldots, ...
1
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1answer
173 views

What is a cyclic ideal?

I assume it's an ideal that generates the whole Ring. Also is $(x+y)^2$ a non cyclic ideal of $R=\mathbb{C}[x,y]/(x^2,xy,y^2)$. The notes I'm using don't give a definition of a cyclic ideal, ...
2
votes
1answer
234 views

A ring with IBN which admits a free module with a generator with less elements than a basis

This is a follow-up to this question of mine: Cardinality of a minimal generating set is the cardinality of a basis I observed in a comment to that question that it was, in fact, a duplicate of this ...
0
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0answers
93 views

Find all Gaussian primes in a given range

All $a,b \in \mathbb{Z}$ so that $0\le a \le 4, 1 \le b \le 4$ and $\pi = a+bi$ prime in $\mathbb{Z}[i]$ want to be found. All the possibilities over $\mathbb{Z}[i]$ are : ...
7
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1answer
151 views

Proper ideals generated by central ideals

Let $R$ be a unital ring and denote its center by $Z(R)$. If $I$ is an ideal of $Z(R)$, then the set $RI$ (consisting of finite sums of elements of the form ra where $r\in R$ and $a\in I$) is clearly ...
0
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1answer
146 views

Division in the Ring of Integral Quaternions

Consider the ring of integral quaternions, $\mathbb{I}$, with norm-like function $N(a+bi+cj+dk) = a^2+b^2+c^2+d^2$. Let $z,w \in \mathbb{I}$, with $w \neq 0$. Prove that $\exists q, r \in \mathbb{I}$ ...
4
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1answer
179 views

Definition of graded rings

So, there are two types of definitions of graded rings (I will consider only commutative rings) that I have seen: 1) A ring $R$ is called a graded ring if $R$ has a direct sum decomposition $R = ...
3
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1answer
329 views

the image of $1$ by a homomorphism between unitary rings

let $R$ and $S$ be unitary rings and $\phi:R\rightarrow S$ a ring homomorphism. is the following correct: $\phi(1_R\cdot1_R)=\phi(1_R)\cdot\phi(1_R)$ so $\phi(1_R)(1_S-\phi(1_R))=0_S$ and so ...
0
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1answer
54 views

equivalence factorial ring , p prime divides $a,…,a_{k}$ is equivalent to $a_{1},…,a_{k}$ are not relatively prime

It will be now shown that this when R is a factorial ring, and $a_{1},…,a_{k}$ in R are not all 0. Then $a_{1},…,a_{k}$ not relatively prime $\Leftrightarrow$ there exists a prime $p$ so that ...
11
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2answers
2k views

Prove that a UFD $R$ is a PID if and only if every nonzero prime ideal in $R$ is maximal

Prove that a UFD $R$ is a PID if and only if every nonzero prime ideal in $R$ is maximal. The forward direction is standard, and the reverse direction is giving me trouble. In particular, I can prove ...
0
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2answers
35 views

For a polynomial $A$, there are no chains of ideals containing $AR$ of length larger than the degree of $A$

Let $K$ be a field, let $R=K[x]$, and let $A\ne 0 \in R$. If $m > \deg(A) $, it is to show that there are no ideals $I_{1}=AR, I_{2},\ldots,I_{m}$ of $R$, all different from each other, such that ...
0
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1answer
171 views

Correct proof that irreducible in a factorial ring is always prime

It will be shown now that every irreducible element in a factorial ring is also prime. Let R be a factorial ring. then let $a\in R$ be irreducible and $x,y \in R$ so that $a| xy$. It is now to ...
4
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1answer
106 views

no prime numbers in a disc in $\mathbb{C}$ with radius R in $\mathbb{Z}[i]$

Show that there is a disc in $\mathbb{C}$ with radius R , so that no primes of $\mathbb{Z}[i]$ are contained in the disc. I was thinking of taking a disc which which does not touch (0,0), for ...
0
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1answer
267 views

Is every subring of a field over $\mathbb{Q}$ a finitely generated $\mathbb{Z}$-module?

Let $R$ be a subring of the field $K$ which is a finite extension of $\mathbb{Q}$. Is $R$ a finitely generated $\mathbb{Z}$-module? I want to say this is true, because we can think of $K$ as a ...
2
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3answers
457 views

How to show associativity of multiplication of polynomials in $R[x]$, where $R$ is a commutative, associative ring

Suppose that we have a commutative, associative ring $R$ which we use to generate the polynomial $R[x]$. Then $p(x) \in R[x]$ is of the form $p(x) = \sum_{i=0}^n a_i x^i$ for some $n \in \mathbb{N}$. ...
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1answer
3k views

Irreducible but not prime in $\mathbb{Z}[\sqrt{-5}] $

It is to show that $2,3, 1-\sqrt{-5}, 1+\sqrt{-5}$ are irreducible over $\mathbb{Z}[\sqrt{-5}]$ but not primes and that 1 and -1 are the only units. Let $N$ be the norm map into $\mathbb{Z}$ and let ...
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0answers
148 views

Is “P is a left primitive ideal” implies that there is a left maximal ideal…?

By definition, a primitive ideal $P$ exists if there is a simple $R$-module $S$ such that $Ann(S)$=$P$. I saw another statement as follows: "$P$ is a primitive ideal of a ring if there is a left ...
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3answers
696 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?
4
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2answers
179 views

Projective modules over $k[X,Y]/(X^3,Y^5)$

I'm searching for an example of a module, that is not projective for $k[X,Y]/(X^3,Y^5)$, but projective for the two subalgebras $k[X]/(X^3)$ and $k[Y]/(Y^5)$. (I don't think it is relevant, but in ...
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1answer
76 views

Proving $\mathrm{ann}_{1}(x)=A_{1} \partial^2+ A_{1}(x \partial -1)$

I don't understand the proof of $$\mathrm{ann}_1(x) = A_1\partial^2 + A_1(x\partial - 1),$$ where $\mathrm{ann}_r(S)=\{r \in R :rm=0, \forall m \in S\}$. $A_1$ is a left module. The ring is ...
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3answers
103 views

For $k\geq 2$ and $m_1,\ldots,m_k \in \mathbb{N}$ with $\gcd(m_i,m_j) = 1$ for $i\neq j$, show that $f(x)$ is a ring homomorphism

Let $k\ge 2$ and $m_{1},…,m_{k} \in \mathbb{N}$ with $\gcd(m_{i},m_{j}) = 1$ for all $i\ne j$. Show that $f(x) = (x,…,x)$ defines a ring homomorphism $f: \mathbb{Z}/m\mathbb{Z} \rightarrow ...
3
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1answer
170 views

every $I\!\trianglelefteq\!R$ is free $\Longleftrightarrow$ $R$ is a PID

In a discussion on MO, I found someone claiming the following: Proposition: For a commutative unital ring $R$, the following are equivalent: (i) every submodule of a free $R$-module is free; (ii) ...
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1answer
306 views

Principal ideal ring

Does there exist a ring which is not a principal ideal ring and which has exactly six different ideals? (For me a ring is commutative with a unit element.) I can show that any ring having at most ...
6
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1answer
109 views

Finiteness of the Witt ring

Is there some slick proof of the fact that for a field $F$, the Witt ring $W(F)$ is finite if and only if $-1$ is a sum of squares and $F^\times/F^{\times 2}$ is finite?
3
votes
1answer
391 views

Consequence of Nakayama Lemma's to Local Rings

I have read that Nakayama's Lemma has a nice consequence for local rings. If $R$ is a local ring with a finitely generated module $M$ and maximal ideal $\mathfrak{m}$, then we see $M/\mathfrak{m}M$ is ...
3
votes
2answers
265 views

Do we get this lucky property in Noetherian rings?

One definition of a Noetherian ring is that every ideal is finitely generated. If $R$ is a Noetherian ring and $I$ and idempotent ideal, that is, $I^2=I$, would it be too far-fetched to guess that ...
1
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0answers
124 views

What's the easiest way to calculate a power series/polynomial modulo a prime?

I'm learning about polynomials stored in a closed form that resembles a generating function or power series. Generally speaking, I have fractions of polynomials, with one example being ...
5
votes
3answers
734 views

Units in quotient ring of $\mathbb Z[X]$

An exercise from Dummit & Foote: Determine the units of the ring $A = \mathbb{Z}[X]/(X^{3})$ and the structure of the unit group $A^{\times}$. Help would be great. Thanks!