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

learn more… | top users | synonyms (2)

0
votes
1answer
20 views

quick question on ascending chain condition for rings

I know that if $R$ is a commutative ring with an identity in which every ideal if finitely generated then it satisfies the ascending chain condition. Just wondering if the converse is also true?
-4
votes
0answers
29 views

Integral domain and ideal

Let $(R,+,*)$ be an integral domain, and $S$ a normal subgroup of $R$, show then that the set $P=\left\{r \in R\ |\ \exists x \in R\ \backslash\ \{0\},\ r*x \in S\right\}$ is the domain of the ideal ...
0
votes
0answers
20 views

Factoring polynomials when roots are external to the ring

I shall avoid maths script since I'm typing on a mobile, anyway I think I can do without. I have a question about factoring polynomials over a ring. Let's call R the ring in question. It is clear to ...
1
vote
1answer
44 views

If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is $R$ finite?

If $R$ is an integral domain with unity having only finitely many subdomains (not necessarily with unity), then is it true that $R$ is finite ? (I know that there are infinite domains with unity, ...
2
votes
3answers
30 views

$D$ be a UFD , if an element of $D$ is not a square in $D$ then is it true that , that element is not a square in the fraction field of $D$?

Let $D$ be a UFD , , let $F$ be the field of fractions of $D$ , let $a \in D$ be such that $x^2 \ne a , \forall x \in D$ , then is it true that $x^2\ne a ,\forall x \in F$ ? (This problem is ...
-1
votes
2answers
34 views

Polynomial ring, ideals and Spec

Morning everyone, I want some hint about this. i) Determine all ideals of $\frac{\Bbb{R[X]}}{<X^3-1>}$ where $R$ is real set ii)Is $\frac{R[X]}{<X^3-1>}$ integral Domain iii)...
1
vote
1answer
36 views

Delooping of a ring?

I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object ...
0
votes
1answer
31 views

Rings with exactly two zero-divisors [on hold]

Why rings with only two zero-divisors are $Z_4$ and $Z_2[X]/(X^2)$?
0
votes
2answers
61 views

Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R.

Indicate True/False Let $R$ be a ring with unity. Consider $X^2 - 1$ $\in$ $R[X]$. Then $X^2 - 1$ has at most two roots in R. I need a hint to solve this problem. I have tried some common rings ...
1
vote
1answer
60 views

Universal property, localization of rings and modules, and initial element in a category

Sorry for the confusing title. I just started learning category theory and am very confused about the concept "universal property". I am not even sure whether my "proof" is a proof or is just a ...
3
votes
2answers
56 views

Polynomial ring with arbitrarily many variables in ZF

For a given field $k$ and a set $X$ we want to define the ring $k[X]$ of polynomials with $X$ as the set of variables. We do not assume $X$ to be finite. And we want to do this without employing axiom ...
0
votes
0answers
19 views

An example of a module that have no supplement.

We see that if R/J is not coclosed coprojective and J has a supplement then R/J is projective. Now we are looking for an example J has no supplement and also R/J is not coclosed coprojective. But we ...
0
votes
1answer
22 views

Showing that $\mathrm{Rad}((0)) ≠ (0)$ implies $R^\times \subsetneq R[X]^\times$

Let $R$ be a commutative ring with $1$, and let $I ≤ R$ be an ideal. We call $\mathrm{Rad}(I) := \{r \in R: \exists n \in \mathbb{N}_0: r^n \in I\}$ the radical of $I$. I now want to show that if $\...
1
vote
1answer
42 views

$k[x,y,z]/(y-x^2,z-x^3)\cong k[x]$, where $k$ is a field

This is generalizing from a previous question, which asks to prove that $k[x,y]/(y-x^2)\cong k[x]$. The way I proved that was by using the homomorphism $\phi:k[x,y]/(y-x^2)\to k[x]$, $\phi(\overline{f(...
1
vote
2answers
76 views

Show that $R$ is a field

Let $R$ be a commutative ring with unit. If $R\neq 0$ such that each finitely generated $R$-module is free then $R$ is a field. In my notes there is the following proof: We need to show that ...
0
votes
1answer
56 views

Is it true that $\mathbb{Q}[x,y]/(xy^2-1)\cong\mathbb{Q}(x)[y]/(y^2-\frac 1x)$? [on hold]

I need to show that $(xy^2-1)$ is prime in $\mathbb{Q}[x,y]$ and I tried to consider the isomorphism $$\mathbb{Q}[x,y]/(xy^2-1)\cong\mathbb{Q}(x)[y]/(y^2-\frac 1x).$$ Does it hold? Thank you.
0
votes
1answer
24 views

In an $\Bbb{N}$-graded domain $A$, units are homogeneous

Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish ...
0
votes
0answers
28 views

Describe the prime elements of the ring $\mathbb Z[\sqrt{-2}]$ [duplicate]

I have a ring $\mathbb Z[\sqrt{-2}]$ and I need to describe all the prime numbers of that ring. How I can do that? Thank you
0
votes
2answers
81 views

Polynomial with infinite roots

I started Ring theory recently and I came across this statement while reading polynomial rings.. If $F$ is an infinite field and let $f(x)\in F[x]$ . If $f(a)=0$ for infinitely many elements $a$ ...
0
votes
1answer
43 views

If for all $a\in A$, $a\neq 0$ exists $b \in A$ such that $ab\neq 0$ then prove that $A\cong T_A$.

$A$ is a commutative ring with identity, and $T_A =\{ T_a \mid a\in A\}$, where $T_a=ax$ for all $x\in A$. If for all $a\in A$, $a\neq 0$ exists $b \in A$ such that $ab\neq 0$ then prove that $A\...
12
votes
1answer
106 views

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$?

Does there exist a polynomial $p(x) \in \mathbb C[x]$ such that $p(x) \notin \mathbb R[x]$ and $p(x)p(-x)=p(x^2)$ ? I have noticed that if $a_n$ is the leading co-efficient of $p(x)$ then $a_n=(-1)^n ...
0
votes
1answer
52 views

Each automorphism is of that form

Let $R$ be a commutative ring and $c,b\in R$ with $c$ invertible. The correspondence $x\rightarrow cx+b$ defines an unique automophism of $R[x]$ that is the identity in $R$. If $D$ is an integral ...
2
votes
1answer
36 views

If $a$ and $b$ are elements in a ring with $a^n=b^n$ and $a^m=b^m$ then $a=b$

I was doing the first exercises from the book Exercises in Basic Ring Theory by G. Călugărescu and P. Hamburg and I found one whose solution isn't quite clear to me. Ex. 1.4 If $a$, $b$ are ...
1
vote
1answer
45 views

Exercise on the ring $\mathbb Z \times \mathbb Z$ and its quotient with an ideal

Let $A = \mathbb Z \times \mathbb Z$ a ring, where operations are defined elementwise. a) Prove that the ideal $I$ generated by $x = (4,6)$ is not maximal. b) Find in $A$ (if it exists) an ...
7
votes
0answers
42 views

Maximal ideals in the ring of measurable functions

The $R$ ring of continuous functions from $[0,1]$ to $\mathbb{R}$ has a property that its maximal look like a subset of $R$ consisting of those functions which vanish at a common single point in $[0,...
-1
votes
0answers
31 views

Principal Ideal in a polynomial ring [duplicate]

Let $K[x]$ be a polynomial ring. If I am given two polynomials $P_1$ and $P_2$, and if I find the generator of the ideal of those two polynomials, How can I tell whether or not that ideal is principal?...
2
votes
1answer
23 views

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$, does that mean that $s\not\mid r$ in $R$?

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$ and $r,s\in R\setminus\{0\}$, does that mean that $s\not\mid r$ in $R$? I was thinking for example in $\Bbb{Z}$, ...
0
votes
2answers
88 views

Is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$?

I came over a question in ring theory which I am not being able to proceed upon: When is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$, where $m, n \in \mathbb{N}$? I know that to ...
1
vote
1answer
100 views

Prove that $\Bbb{R}[\cos(\theta),\sin(\theta)]\cong\Bbb{R}[x,y]/(1-x^2-y^2)$ [duplicate]

More precisely, given the ring homomorphism $\phi:\Bbb{R}[x,y]\to\Bbb{R}^\Bbb{R}$, with $\phi(f(x,y)):\Bbb{R}\to\Bbb{R},\,\,\phi(f(x,y))(\theta)=f(\cos(\theta),\sin(\theta))$, where $\Bbb{R}[x,y]$ is ...
0
votes
1answer
23 views

Is it true that the only regular elements in $Z_m$ are invertible ones?

I have this doubt. In a unitary and commutative ring $$Z_m = \{[0]_m, [1]_m,\ ...\ ,\ [m - 1]_m\}$$ There are only two "kind" of elements: invertible and zero divisors. Is it true to say that the ...
0
votes
1answer
19 views

the field $Fr (A [X])$ and $ Fr (A) (X)$ are the same

Let $A$ be an unitary integral domain , and let $Fr (A)$ its fractionary field; this field is determined (to isomorphism field) by the following universal property: a) the ring $A$ is injected by a ...
-3
votes
1answer
24 views

Why does this hold in these cases? [closed]

Let $R$ be a U.F.D. and $0\neq d\in R$. If $d\notin U(R)$ do we have that $d=a_1^{k_1}\dots a_r^{k_r}$ with $a_i$ irreducible? If $d\in U(R)$ why does it hold $(d)=R=(1)$ ?
1
vote
1answer
30 views

A question on part of the proof of the theorem that if $R$ is a UFD then $R[x]$ is a UFD as well.

I have a question regarding a proof in Peter Falb's Methods of Algebraic Geometry in Control Theory, volume I, for the claim in the title. On pages 16-17 he proves the property (ii) of UFDs that the ...
1
vote
1answer
21 views

Name for submodule killed by a right ideal

Let $\mathfrak a$ be a right ideal in a ring $R$. The set $N=\{m\in M: \mathfrak am=\mathfrak 0\}$ is a submodule of the left $R-$module $M$: If $m,n\in N$, $a\in \mathfrak a$, then $a(m-n) = am-an =...
2
votes
2answers
262 views

Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
1
vote
1answer
41 views

Quotient ring with reducible polynomial

Let $S = \mathbb{R}[x]/(x^2+1)^2).$ The first goal is to show that there exist exactly two homomorphisms $ \pi\colon S\to \mathbb{C} $ such that $\pi|_{\mathbb{R}} = \text{id}_{\mathbb{R}}.$ I know ...
0
votes
0answers
28 views

Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
1
vote
2answers
115 views

Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
1
vote
1answer
32 views

$\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors

I want to show that $\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors: given a chain $a_1,a_2,\dots,a_n,\dots$ and $a_{n+1}\mid a_n$ for any $n\in \mathbb{N}$, then there is ...
0
votes
1answer
47 views

$I$ is the maximal left ideal

Let $R$ be a ring and $I\subseteq R$ the unique maximal right ideal of $R$. I have shown that $I$ is an ideal and that each element $a\in R-I$ is invertible. I want to show that $I$ is the unique ...
-2
votes
2answers
64 views

Is $a$ invertible?

We have that $R$ is a ring. Suppose that $Ra=R$ and $bR=R$, for $a,b\in R$. Then we have that there is $x\in R$ such that $ab=1$ and $bx=1$. Does it follow from that that $a$ is invertible?
3
votes
0answers
77 views

Simple examples of rings from topology

The ring $C([0,1],\mathbb{R})$ of continuous functions from $[0,1]$ to $\mathbb{R}$ is an interesting example of ring due to its some interesting property (namely, structure of maximal ideals). The ...
0
votes
0answers
44 views

Field of fractions of ring F[x] [closed]

Let $F$ be a commutative ring without zero divisors and $Q$ its field of fractions. Prove that field $Q(x)$ is a field of fractions of ring $F[x]$ Thanks for any help.
1
vote
0answers
59 views

Which functions $\mathbb{Z} \rightarrow \mathbb{Z}$ are 'totally compatible'?

Definition 0. For each integer $k$ and each function $f : \mathbb{Z} \rightarrow \mathbb{Z}$, lets define that $f$ is $k$-compatible iff there exists a function $g : \mathbb{Z}/k\mathbb{Z} \rightarrow ...
3
votes
0answers
69 views

When does $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges?

We know $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[i]$ (because that sum is greater that $\sum_{p \equiv 3 \mod 4} \frac{1}{p}$, which ...
0
votes
1answer
29 views

If $a$ is algebraic and $f\colon\mathbb{Q}[x]\to\mathbb{C}$ where $f(g(x))=g(a)$, prove that $\ker(f)$ is a maximal ideal of $\mathbb{Q}[x]$

If $a$ is algebraic, then a polynomial $p(x)$ in $\ker(f)$ is irreducible iff it generates $ker(f)$. For an ideal $I$ in $Q[x]$ containing $\ker(f)$, let $p(x)=\ker(f)$ and $q(x)=I$. Then $p$ is ...
1
vote
1answer
29 views

Number of distinct equivalence classes of $\mathbb Z_n$ of the “ associate ” equivalence relation

Define an equivalence relation on $\mathbb Z_n$ as : For $a,b \in \mathbb Z_n $ , $a\sim b$ iff $\exists k \in U_n=\mathbb Z_n^{\times}$ such that $a=kb$ (i.e. $a,b$ are related if they are "...
0
votes
3answers
34 views

If $a$ is algebraic, prove that there is a minimal polynomial $p(x)$ in $Q[x]$ such $p(a)$ = $0$.

If $f_a$: $Q[x]$ -> $C$ is the evaluation at $a$ map, then a polynomial $q(x)$ in $ker(f_a)$ is irreducible iff it generates $ker(f_a)$. Let $ker(f_a)$ = $h(x)$ so that $h(x)$ is irreducible and $f_a(...