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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0
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1answer
33 views

Proof that primary submodules of $R$ are primary ideals of $R$

I want to prove this: Let $R$ be a commutative ring with identity. If $Q$ is a primary submodule of $R$ (as an $R$-module), then $Q$ is a primary ideal. $Q$ is a primary submodule of $R$ if $r \in R$...
4
votes
2answers
92 views

Element of a ring without unity which divides every other element

Question. Is there an example of a ring $R$ (commutative or not) without unity and an element $x \in R$ such that for every $y \in R$ there exists a $z \in R$ such that $y = x z$? In other words, is ...
-1
votes
1answer
36 views

Let $f\in \mathbb{Z}_n [[X]] $ be a non-zero power series all of whose coefficients are nilpotent. Show that $f$ is nilpotent. [closed]

Let $f\in \mathbb{Z}_n [[X]] $ be a non-zero power series all of whose coefficients are nilpotent. Show that $f$ is nilpotent. I don't know how to proceed.. Thanks for any help!
1
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0answers
24 views

On a stronger property than being an Armendariz ring

A ring $R$ is said to be Armendariz if $f(x), g(x) \in R[x]$ are such that $f(x)g(x) = 0$, where $f(x) = a_nx^n + \dots a_0, g(x) = b_mx^m + \dots + b_0$, then $a_ib_j=0$ for all $i,j$. In other ...
1
vote
0answers
68 views

Do we have that $y\in R\setminus R^\times$, such that $y^n\mid x$ for every $n\in \mathbb N\implies x=0$?

Is it true that if $R$ is an integral domain with $1$, and for $x\in R$ there exists $y\in R\setminus R^\times$ such that $y^n$ divides $x$ for every $n\in \mathbb{N}$, then $x=0$? I think this is ...
1
vote
1answer
48 views

How many ways can $\mathbb{Z}/5\mathbb{Z}$ be given the structure of a module over Gaussian integers?

As the title says, the question is: In how many ways can the additive group $\Bbb{Z}/5\Bbb{Z}$ be given the structure of a module over Gauss integers? My attempt: Any $\Bbb{Z}[i]$-module M is an ...
-1
votes
1answer
48 views

Is the set of all 2x2 matrices a ring with zero divisors? [closed]

Let M be the set of all $2\times2$ matrices with real entries. Under matrix addition and multiplication, is it a ring with zero divisor? Verify whether the above is true or false with justification. ...
0
votes
2answers
25 views

In any ring R, show that ab is nilpotent iff ba is nilpotent. Can one say the same for zero divisors?

I tried it this way... $(ab)^{n}=o$ then $a(ba)^{n-1}b=0$ but then we can't conclude anything from this as a and b may not have inverses.. Thanks for any help!!
1
vote
1answer
39 views

subring of a finite ring that is division ring

Is there any noncommutative ring of finite order whose subring is a division ring? I started with taking matrix ring but not been able to get the answer yet.
2
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0answers
54 views

Novikov Ring Proof

I have a four part question involving the Novikov ring. For part (1), I have: $(\Sigma_{\gamma\in \tau}a_{\gamma}t^\gamma) +_{N} (\Sigma_{\gamma\in \tau}b_{\gamma}t^\gamma) =\Sigma_{\gamma\in \...
5
votes
2answers
162 views

Relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring for dummies?

As the question title suggests, what is an explanation for dummies of the relationship between Fourier coefficients, eigenvalues, and the spectrum of a ring?
1
vote
0answers
40 views

Prove that $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain [duplicate]

Question: Let $\alpha = \sqrt{-2}$ and let $R = \{ x + y \alpha | x, y \in \mathbb{Z}\}$ Show that $R$ is a Euclidean domain. My attempt I'm not entirely sure where to start so I have tried to ...
0
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0answers
15 views

In a DVR, why does $u=f(t,u)$, with $f$ a homogeneous polynomial of degree $3$ and $t$ a uniformizer, imply $\nu(u)=3$?

In this answer by Georges Elencwajg, it is stated that $$u=t^3-\dots-e_1e_2e_3u^3=\text{a homogeneous polynomial of degree $3$ in t,u}\quad(\ast)$$ [...] Now in the local ring $\mathcal O_{...
1
vote
4answers
60 views

Show that $\langle3\rangle$ is a maximal ideal in $\mathbb{Z}[i]$ [closed]

Equivalently how can I show that $\mathbb{Z}[i]/\langle 3\rangle$ is a field?
1
vote
2answers
25 views

The ring of all continuously n-times differentiable functions is not an integral domain for all n?

Let $\mathbb{Z^+}$ be the set of all non-negative integers. For each $n\in \mathbb{Z^+}$ let $S_{n} =C^{n}([0,1], \mathbb{R})$ be the ring of all continuously n-times differentiable functions where $...
1
vote
1answer
19 views

Euclidean Domain, Associates

Let $R$ be an Euclidean Domain and $f$ be the Euclidean valuation on $R$. Show that if $a$ and $b$ are associates in $R$, then $f (a) = f (b)$. What I tried: I know that $f$ is Euclidean Valuation so,...
0
votes
1answer
53 views

Prove that P is prime ideal of R?

$R$ is a commutative ring and $1\in R$. Let $I$ be an ideal of $R$ and $P$ be a prime ideal of $I$. Then show that $P$ is prime ideal of $R$. I know how to prove that P is ideal of R. Suppose that $...
0
votes
1answer
42 views

extension of derivation of algebras

I am studying extension of derivations, but I am confused of some notations and maybe symbols! For some more details, I recall a theorem. We have the following theorem : Theorem: Let $A$ be an ...
1
vote
1answer
49 views

Which of the following rings is isomorphic with $ \Bbb{Z}_2 [x]\big/ \langle x^2 \rangle $

Which of the following ring is isomorphic with $\Bbb{Z}_2 [x] \big/ \langle x^2 \rangle $: $ \Bbb{Z}_4 $ $ \Bbb{Z}_2 \times \Bbb Z_2 $ We know that cardinality of the ring $ \left|\Bbb{Z}_m [...
2
votes
1answer
63 views

Is $\mathbb{Q}(\sqrt{2})$ a Euclidean domain? [duplicate]

How do I prove that $\mathbb{Q}(\sqrt{2})$ is or isn't a Euclidean domain? So if $F$ is a field, then $F[X]$ is a Euclidean domain. I don't see why this means that $\mathbb{Q}(\sqrt{2})$ is a ...
0
votes
2answers
24 views

Let $R$ be an integral ring and let $F$ be a field contained in $R$. $R$ is a vector space over $F$. Show that $R$ is a field.

Let $R$ be an integral ring (has no zero divisors) and let $F$ be a field contained in $R$. Supposse that $R$ is a vector space over $F$. Show that $R$ is a field. This promem seems very easy, since ...
4
votes
0answers
44 views

Left ideals of $M_2(K)$ with $K$ a field

Is it true that the only proper left ideals of $M_2(K)$, the ring of the matrices whose coefficients are in a field $K$, are $$ \left\{\begin{pmatrix}ah & ak \\ bh & bk\end{pmatrix}: a,b \in K\...
-3
votes
1answer
44 views

Is it true that $I+J\subseteq I\cap J$ [closed]

Let $R$ be a commutative ring and $I,J$ are ideals in $R$. Is it true that $$I+J\subseteq I\cap J?$$ I know that $I+J =\{x+y|x\in I, y\in J\}$.
0
votes
2answers
58 views

Understanding divison by monic polynomial in $R[x]$ where $R$ is an arbitrary ring

I read "Algebra: Chapter 0" by P.Aluffi. I encountered a topic where it says you can divide any polynomial in $R[x]$($R$ is any ring) by a monic polynomial(that is, a polynomial of the form $x^d + \...
4
votes
1answer
37 views

Maximal ideals of $C\big((0,1)\big)$

We know that all the maximal ideals of $C([0,1])$ is in the following form $$ M_a=\{f\in C[0,1]\ :\ f(a)=0 \}\ \text{for some } a\in [0,1] $$ But suppose that if we will replace our domain to compact ...
1
vote
1answer
88 views

Number of distinct prime ideals in $\mathbb {Q}[x]/\langle x^m-1 \rangle$ [closed]

Number of prime ideals in quotient ring obtained by $\mathbb {Q}[x]/\langle x^m-1 \rangle$ is ...?
1
vote
1answer
24 views

Essential Prime Ideal

I search for an example of a commutative ring $R$ with unity having a prime ideal $P$ and some element $r\in R$ such that the annihilator of $r$ is both contained in $P$ and essential in $R$. By ...
0
votes
1answer
62 views

About commutative ring with identity

Let $R$ be an infinite commutative ring. Which of following options is false? Center of $M_2(R×R)$ is nontrivial. $ M_2(R×R) \cong M_2(R)×M_2(R)$ The number of units in $M_2(R ×R)$ is infinite. The ...
-1
votes
1answer
30 views

Ideal which is maximal with respect to the property that it does not contain the set S

Question Let $R$ be a ring and let $S$ be a subset of $R$. Prove that there exists an ideal that is maximal by inclusion with respect to the property that it does not contain the set $S$. I ...
9
votes
3answers
384 views

Defining binomial coefficients over rings in general

I'm looking over my notes, and we proved that the binomial theorem $$(a + b)^n = \sum_{k = 0}^{n} {n \choose k}a^k b^{n-k}$$ holds on all commutative rings. However, I am confused on how we are ...
-1
votes
1answer
24 views

UFDs and reduced quotients

Let $A$ be a UFD and let $x\in A$ an element. I don't understand why the following claim is true: The quotient $A/(x)$ is reduced if and only if $x$ is a product of distinct primes. Can you suggest ...
1
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2answers
33 views

Why Is $I$ A Maximal Ideal If $R/I$ Is A Field?

How can I prove that if $R/I$ is a field then $I$ is a maximal ideal? In my book it says that this is a corollary of the following theorem: If $φ:\mathbb{F}→S$ is a non trivial homomorphism and $\...
2
votes
1answer
29 views

For $\mathbb{Z}[\sqrt{-10}]$ and an ideal $I=(2,\sqrt{-10})$, find $n>0$ s.t. $I^n$ is a principal ideal.

I can show that $I$ itself is not a principal ideal by using the multiplicative norm $N(a+b\sqrt{-10}) = a^2 + 10 b^2$ using the following: Suppose that $I=(\alpha )$. Then $N(\alpha) | 2$ and $N(\...
2
votes
1answer
45 views

Why is it necessary to show that the set R is closed under multiplication in order to prove that (R,+,*) is a Ring.

I am currently reading John B Fraleigh's book A First Course in Abstract Algebra. The author in chapter 23, insists on the point that it is necessary for one to show that the R is closed under ...
5
votes
2answers
56 views

How to construct rings with a given class number?

Hi I was learning about class number and I was wondering if it is known how to construct rings for any specific class number.
0
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1answer
35 views

Conjecture about the elements of quotients of polynomial rings over commutative rings by principal ideals

Let $R$ be a commutative ring with unity. For any polynomial $p(x) \in R[x]$ is there an element $a$ of $R[x]/ \langle p(x) \rangle$, $a \not \in R$ such that $a^n \in R$, where $n$ is the degree of $...
3
votes
1answer
73 views

Example of non-principal ideal of the quotient ring $R[x,y,z]/(xyz-1)$

Suppose $R$ is an integral domain, $R[x,y,z]$ is polynomial ring over $R$, and $$Q=R[x,y,z]/(xyz-1)$$ is a quotient ring. How to prove, that $Q$ is not principal ideal ring? I was trying to compose an ...
1
vote
1answer
34 views

Direct sum decompositions over rings.

Let $R$ be a ring and let $I$ be one of its nonzero left-ideals. Is it true that $R \cong I \oplus R\big/I$? Notice that we can consider $R$ to be an left $R$ module and $I$ one of its $R$-submodules....
2
votes
1answer
33 views

Are Noetherian Rings Goldie Rings?

We know that a ring $R$ is said to be Right-Goldie if $R$, as a right $R$ module, satisfies: (i) $R$ has finite uniform dimension; (ii) every ascending chain of right annihilators of $R$ terminate. ...
0
votes
1answer
35 views

$p_1q_1q_2$ and $p_1p_2q_1$ do not have a greatest common divisor

Question Let $p_1,p_2,q_1,q_2$ be irreducible elements in integral domain $R$ such that none are associates to any of the others and $p_1p_2=q_1q_2$. Prove that $p_1q_1q_2$ and $p_1p_2q_1$ do ...
0
votes
1answer
18 views

Some equivalences for Ideals of the ring of real valued functions

Let $I_{N}=\{f\in\Omega(\Bbb R,\Bbb R)|\forall x \in N\subset \Bbb R: f(x)=0\}$, where $\Omega(\Bbb R,\Bbb R)=\{f:\Bbb R \to \Bbb R|f\space is \space a\space function\}$, then the following statements ...
0
votes
1answer
31 views

quotient of P.I.D by a prime power a P.I.D? [closed]

If $R$ is a P.I.D. and $p\in R$ is prime, is it the case that $R/<p^k>$ will be a P.I.D for all k? If so how would one show this?
1
vote
1answer
10 views

if an element $q_1$ of a principal ideal domain is irreducible, then the ideal $q_1 A$ is a maximal ideal

In my lecture, my professor wrote within a proof: $q_1$ in the principal ideal domain $A$ is irreducible, therefore, the ideal $q_1A$ is a maximal ideal. I don't understand how that's true. I tried ...
1
vote
1answer
24 views

Structure of the proof of the fundamental theorem of arithmetics

Fundamental theorem of arithmetics: A principal ideal domain is factorial. i.e: Any non zero element of a principal ideal domain can be decomposed in a unique product of irreductibles. The structure ...
3
votes
0answers
59 views

Show that $(\mathbb{Z}\oplus\mathbb{Z})[x]$ is not isomorphic to $\mathbb{Z}[x]\oplus\mathbb{Z}[x] $ . [duplicate]

Problem says: Show that $(\mathbb{Z}\oplus\mathbb{Z})[x]$ is not isomorphic to $\mathbb{Z}[x]\oplus\mathbb{Z}[x]$. But consider the map $\varphi:(\mathbb{Z}\oplus\mathbb{Z})[x]\rightarrow\...
1
vote
1answer
108 views

Prove or Disprove: there is only one ring homomorphism $f:\mathbb{R} \rightarrow \mathbb{C}$

I searched a lot but I couldn't solve my problem! I know that $$ f(1) = f(1.1)= f(1).f(1) \Longrightarrow f(1) = 0 \quad or \quad f(1) = 1 $$ I know that if we suppose that $f(1) = 0$ then $f$ is ...
1
vote
1answer
40 views

Question About Irreductibility of an element in a ring

I $A$ is a principal ideal domain and let N the set of non zero elements, that do not have an inverse and that cannot be decomposed in a product of irreductibles. If $a \in N$ Let's create the ...
0
votes
1answer
14 views

Proof of a proposition regarding rings and ideals

I am looking for a proof of the following proposition: If A is a principal ideal domain, let the following increasing sequence of Ideals of A: $I_1⊆I_2⊆I_3⊆...⊆I_m⊆...$ Then there exists $n \geq 1$ ...
3
votes
1answer
24 views

What is known about finite-dimensional non-semisimple associative algebras over $\Bbb C$?

Artin-Wedderburn says the semisimple ones are sums of matrix algebras, but what about the non-semisimple ones? What are some examples?
0
votes
0answers
29 views

Finite module over Noetherian ring faithfully flat?

If I have Noetherian rings $B=A^G\subset A$ (for some action of finite group $G$, maybe not relevant) and $A$ is finite as $B$-module. Is it always true that $A$ is faithfully flat over $B$? EDIT: ...