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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Definition of Irreducible polynomial in terms of the unit of Integral domain.

Let $D$ be an integral domain. A polynomial $f(x)$ from $D[x]$ that is neither the zero polynomial nor a unit in $D[x]$ is said to be irreducible over $D$ if, whenever $f(x)$ is expressed as a product ...
1
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1answer
51 views

Equality of polynomial functions modulo n

Fix positive integers $m$ and $n$. For all polynomial functions $f,g: \mathbb{Z}^m \to \mathbb{Z}$ define the equivalence relation $\sim$ by $$f \sim g \iff \forall x \in \mathbb{Z}^m \ ( \ f(x) \...
5
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1answer
63 views

Is the number of isomorphism classes of quotients of a finite dimensional commutative ring over a field finite?

If $A$ is a finite dimensional unital and commutative algebra over some infinite field $k$, what is the number of isomophism classes of rings of the form $A/I$ where $I$ is a proper ideal of $A$? Is ...
3
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1answer
85 views

Build a reduced ring starting from an ordinary one

This may be easier than I think, but still I can't seem to wrap my head around it. I've learnt that if we take a ring $R$ and quotient it for a (two-sided) ideal $I \subset R$ which is radical, the ...
1
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1answer
35 views

Do quotients by an ideal carry over a ring isomorphism?

Say we have ring $A$ such that $A \cong B/I$ for some ideal $I$ of $B$, and suppose also that $B \cong C$ for some other ring $C$. Does there exist an ideal $J$ of $C$ such that $A \cong C/J$? Also, ...
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3answers
29 views

Example of a communtative ring with two operations where the identity elements are not distinct?

I was introduced to the definition of a field today, as a communtative ring with two operations, like $\Bbb{R} = \langle R, +, -, \cdot, ^{-1} \rangle$ and all the usual axioms; commutativity, ...
2
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1answer
46 views

Showing that $c_{i}\equiv 0\pmod{p}$

Let the numbers $c_{i}$ be defined by the power series identity $$\frac{1+x+x^{2}+\ldots+x^{p-1}}{(1-x)^{p-1}}= 1+c_{1}x+c_{2}x^{2}+\ldots$$ Show that $c_{i}\equiv 0\pmod{p}$ for all $i\geq 1$. $\...
3
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1answer
29 views

Can we construct a ring of order $15$ without identity not isomorphic to $\mathbb{15}$?

I've proved any ring of order $15$ with identity is isomorphic to $\mathbb{Z}_{15}$ but what if the ring is of order $15$ with no identity element ? Can we construct a ring of order $15$ without ...
0
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0answers
38 views

Find an $R$-module homomorphism $f:R\longrightarrow M$ such that $r_0m=0$ implies $r_0f^{-1}(m)=0$ , $(m\in M)$

Let $R$ be a commutative ring with $r_0\in R$ a fixed element, and $M$ be an $R$-module. I search for an $R$-module homomorphism $f:R\longrightarrow M$ such that $r_0m=0$ implies $r_0f^{-1}(m)=0$ , $(...
1
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1answer
44 views

Ideals of Unique Factorization Domain

Let R be a commutative ring with unity such that R[x] is UFD. The ideal (x) of R[x] is denoted by I. Then pick the correct statements from below: 1. I is prime. 2. If I is maximal then R[x] is a PID. ...
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1answer
39 views

Let $R$ be an infinite comutative ring with unity, $M,N$ be $R$-modules, $f:M \to N$ be a surjective module homomorphism; then $|M|=|N ||\ker f|$?

Let $R$ be an infinite commutative ring with unity, $M,N$ be modules over $R$, let $f:M \to N$ be a surjective module homomorphism; then is it true that $|M|=|N || \ker f|$ ($M,N$ are not ...
0
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0answers
36 views

Idempotents in commutative ring of characteristic 2 form a subring

Question: In a commutative ring of characteristic $2$, want to show that the idempotents form a subring. Subring Test is probably the way to go. It is easy to verify the identity element in ...
0
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0answers
51 views

Commutativity theorem in the ring theorem

Suppose that $R$ is a ring such that for any $x\in R$ there exists $1<n(x)\in \mathbb{N}$ such that $x^{n(x)}-x\in Z(R)$. Prove that $R$ is commutative or if it is not commutative, then the ideal ...
5
votes
2answers
64 views

Finite commutative ring with unity and without nilpotent elements

Let $R$ be a commutative ring with unity such that for each $x \in R$ there exists a $n \in \mathbb{N}$, $n>1$, such that $x^n = x$. Then show that $$ R\simeq F_{1}\times F_{2}\times \cdots\times ...
0
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1answer
30 views

Let $R$ be a commutative ring. For $a \in R$ consider the set $(a) = \{r*a | r\in R\}$. Show that $(a) = R$ if a is a unit of $R$ [duplicate]

I tried some values and I think I got the idea. R is the set of values used in the ring. If I use $\mathbb{Z}$, the units are $\{-1,1\}$. If I take 1 for example, I could use it to get every value in $...
0
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1answer
34 views

Implication in local noetherian domain of Krull's dimension 1. [duplicate]

Let $(A,m)$ be a local noetherian domain with Krull dimension $1$. Let $k$ be the field $A/m$. I'm trying to prove that if $m/m^2$ has dimension $1$ as a $k$-vector space, then every ideal $I$ ...
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0answers
39 views

Vector space complement to a multiplicatively closed subspace is an ideal

Let $V$ be a vector space over $\mathbb{C}$ of any dimension and suppose we have an associative multiplication $V \times V \to V$ making $V$ into a commutative ring with unity. Let $V=U \oplus W$ be a ...
-3
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1answer
74 views

Is the notion of Stabilizer of a subset A of a group G absurd?

Is the notion of Stabilizer of a subset,A of a group G is absurd? I don't know whether this makes sense or not,but for curiosity i want to know view of experts. UPDATE i'm dealing with ...
0
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1answer
39 views

Let $M_1$, $M_2$ be Artinian modules over $R$. Then $M_1\times M_2$ is Artinian.

Using exact sequences, it's fairly easy to prove the converse, but I can't figure out how to prove this statement. Suppose we have a descending chain $N_1\supset N_2\supset\cdots$ of $R$-submodules ...
3
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1answer
32 views

$p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $A \to p(A)$ surjective on $M(n,\mathbb R)$?

Let $p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $f: M(n,\mathbb R) \to M(n, \mathbb R)$ defined as $f(A)=p(A) , \forall A \in M(n,\mathbb R)$...
3
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1answer
44 views

$p(x) \in \mathbb R[x]$ be non-constant polynomial , $n>1$ , the function $A \to p(A)$ is surjective on $M(n, \mathbb C)$?

Let $p(x) \in \mathbb R[x]$ be a non-constant polynomial and $n>1$ , then is it true that the function $f:M(n,\mathbb C) \to M(n, \mathbb C)$ defined as $f(A)=p(A) , \forall A \in M(n, \mathbb C)...
4
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1answer
165 views

How do I prove that $\mathbb{Z}[\sqrt{19}]$ is not a UFD?

How do I prove that $\mathbb{Z}[\sqrt{19}]$ is not a UFD? Moreover, how do I prove that $(7,3+\sqrt{19})$ is not a principal ideal? This is the first time I'm dealing with a quadratic integer ring ...
1
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1answer
44 views

Prime elements with the same norm in a Euclidean domain [closed]

Does anybody know whether two prime elements with the same norm in a Euclidean domain are necessarily associated? Any help will be very welcome. UPD 1: It was shown that $2\pm i$ are both primes ...
2
votes
1answer
59 views

what kind of integral domain do the non-infinite surreals form?

https://en.wikipedia.org/wiki/Integral_domain mentions the following chain of inclusions: Principal Ideal domains $\subset$ Unique Factorization domains $\subset$ GCD domains $\subset$ Integrally ...
0
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1answer
29 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?
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0answers
25 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
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1answer
50 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
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3answers
56 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 motivated ...
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2answers
42 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)...
2
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1answer
60 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
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2answers
66 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
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1answer
70 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
83 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 ...
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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
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1answer
23 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
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1answer
50 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
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1answer
89 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
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1answer
28 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 ...
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0answers
30 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
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2answers
88 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
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1answer
44 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\...
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1answer
110 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
55 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
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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 ...
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1answer
46 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
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0answers
46 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,...
2
votes
2answers
44 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
89 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
104 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 ...