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
112 views

Prove that if R and S are nonzero rings then $R\times S$ is never a field.

This question has been asked on here before but I'm looking for some additional insights. The question comes from the section of the Dummit and Foote textbook on the Chinese Remainder Theorem. All of ...
5
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0answers
374 views

Integral homomorphism induces a closed map on spectra

I'm trying to prove the following: Let $f:A\rightarrow B$ be an integral homomorphism (e.g. $B/f(A)$ is a integral extension). Consider $f^{*}: \operatorname{Spec}B \rightarrow ...
2
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1answer
200 views

Characteristic of commutative semisimple rings?

In one of my questions (Structure of the group ring of a direct product?), a statement is made for a commutative semisimple ring of characteristic $p^t, t\geq1$. Now I don't understand why there ...
3
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1answer
104 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
3
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1answer
68 views

Find conditions for there to exist a morphism of rings from $\mathbb{Z}_m$ to $\mathbb{Z}_n$

I know that a necessary and sufficient condition for a ring morphism $\mathbb Z_m\to\mathbb Z_n$ to exist is that $n$ must divide $m$. However, I am having trouble understanding a proof that this ...
8
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1answer
144 views

Ring without $1$ where $\forall r\in R$, $\exists$ $n_r > 1$ such that $r^{n_r} = r$, and not all primes are maximal

On my algebra final exam, there was a problem that essentially asked the following: Let $R$ be a commutative ring such that for all $r\in R$, there exists $n_r\in\Bbb{Z}^{>1}$ with $r^{n_r} = ...
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3answers
377 views

Spectrum of polynomial ring

In M. Reid's Undergraduate Commutative Algebra, the author states that if $k$ is an algebraically closed field then $\operatorname{Spec}{k[x]} = \{0\} \cup k$ (page 21). Is this correct? Instead, ...
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1answer
119 views

Prime ideals in commutative ring

Let $R$ be a commutative ring with $1$ (we take $R$ not to be a field for this post). Must $R$ contain at least one prime ideal that is not maximal? The question is equivalent to the following: For a ...
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1answer
41 views

Sufficiency Condition for PIDs

Letting $R$ be an integral domain, the challenge is to prove that (i) every $a,b∈R$ has a gcd which can be written as an $R$-linear combination of $a$ and $b$, and (ii) for every sequence $a_1,a_2, ...
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2answers
91 views

Are there logarithm functions for arbitrary rings?

The logarithm function for $\mathbb{R}$ satisfies $\log xy = \log x + \log y$ whenever both $\log x$ and $\log y$ are defined. Are their conditions for a ring $R$ which guarantee the existence of a ...
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2answers
109 views

Commutative Ring with Unity and Prime Characteristic

Let $D$ be a integral domain with characteristic $p>0$. It is easy to prove that $p$ is prime. Now, if $R$ is a commutative ring with unity and characteristic $p$, does $p$ be a prime number ...
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1answer
59 views

fraction field of the integral closure

Let $R$ be a domain, $K$ the field of fractions of $R$ and $L$ a finite field extension of $K$. Denote with $R'$ the integral closure of $R$ in $L$. Is it always true that $L$ is the field of ...
2
votes
1answer
111 views

The set of zero-square elements in a commutative ring

Let $R$ be a commutative ring and let $I:=\left\{x \in R : x^2=0 \right\}$. Prove that $I$ is an ideal in $R$ or give a counterexample. Remark: This is problem 3B in the January 2003 Algebra ...
2
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1answer
184 views

Annihilator of an element of a left module

How to show that the annihilator of an element of a left module is a left ideal but not necessarily a two-sided ideal. When does this become an ideal?
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0answers
73 views

Find integer solutions to $x^2+xy+11y^2=p$ using Ring identities

Let $\theta = (1+\sqrt{-43})/2$ and consider $\mathbb{Z}[\theta]$, a principal ideal domain, with the multiplicative map $\psi (a+b\theta)=a^2+ab+11b^2$. Show there exists an integer solution to ...
2
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1answer
77 views

Is localization of a prime ideal still a prime ideal?

Im still new to the topic so this question might seem trivial. But I hope if someone can help explaining to me if a prime ideal $P$ of a domain $A$ is still a prime ideal $P_s$ in the localization ...
6
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1answer
179 views

Abstract algebra T/F questions.

This is from our review and my study group is wondering if we can get some feedback on our progress: $1$. The symmetric group $S_3$ only has two proper normal subgroups. True, because $e \subset ...
10
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2answers
460 views

What is a primary decomposition of the ideal $I = \langle xy, x - yz \rangle$?

Given the ring $k[x,y,z]$, where $k$ is a field, and an ideal $I=(xy,x-yz)$, find the primary decomposition of $I$. I tried to draw the graph of the variety of $I$ and get a decomposition of ...
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0answers
400 views

Classification of local Artin (commutative) rings which are finite over an algebraically closed field.

A result in deformation theory states that if every morphism $Y=\operatorname{Spec}(A)\rightarrow X$ where $A$ is a local Artin ring finite over $k$ can be extended to every $Y'\supset Y$ where $Y'$ ...
5
votes
1answer
56 views

What's the difference of naming a polynomial ring as $\mathbb{C}\{ x,y\}$ and $\mathbb{C} [x,y]$?

I sometimes see both notations and I am led (maybe misled) to believe that they are the same thing. What is the formal difference between both of them? Or there isn't any?
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1answer
131 views

When does “second annihilator” of a (principal) ideal equal the ideal itself

Suppose that $R$ is a (local) ring and $r\in R$. When do the equations $\operatorname{Ann}_R(\operatorname{Ann}_R(r))=Rr$ or $\sqrt{\operatorname{Ann}_R(\operatorname{Ann}_R(r))}=\sqrt{Rr}$ hold? I ...
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2answers
384 views

Group of invertible elements

Let R be a ring with unity. How can I prove that group of invertible elements of R is never of order 5? My teacher told me and my colleagues that problem is very hard to solve. I would be glad if ...
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1answer
59 views

Splitting field questions.

$K$ is a field and $f \in K[x]$ with splitting field $L$. Show that $[L:K] \le n!$, where $n$ is the degree of $f$. $f \in \mathbb{Q}[x]$ is a cubic polynomial and $K$ is its splitting field. What ...
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2answers
93 views

Field extension question.

F, K, L are fields. F is extended to K, and K is extended to L. Show that $[L:F]=[L:K] \cdot [K:F]$. Also consider the extension from $\mathbb{Q}$ to $\mathbb{Q}(\alpha)$ where $\alpha = \sqrt{3} + ...
5
votes
1answer
129 views

GCD in a subring is GCD in a bigger ring

Let $R$ be a UFD which is a subring of an integral domain $S$. If $r_1$ and $r_2$ are two nonzero elements of $R$ with GCD $d$, is it true that $d$ is also a GCD of $r_1$ and $r_2$ in $S$? I know ...
4
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1answer
45 views

Question regarding nilpotent ideals of a ring.

I am working on the following: An ideal $N$ is called nilpotent if $N^n$ is the zero ideal for some $n\geq1$. Prove that the ideal $p\mathbb{Z}/p^m\mathbb{Z}$ is a nilpotent ideal in the ring ...
6
votes
1answer
104 views

What are the ring morphisms $\mathbb{Q}[[X]]\to R$ for a ring $R$?

If $\mathbb{Q}[[X]]$ is the ring of power series over a field $F$, then can we describe ring morphisms $\mathbb{Q}[[X]] \to R$ for rings $R$ in simple terms? I am guessing that a "substitution ...
2
votes
2answers
123 views

$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ exact, $M''$ flat. Why is $M$ flat $\Leftrightarrow M'$ flat?

Let $A$ be a commutative ring with identity, and let \begin{align} 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 \end{align} be an exact sequence of $A$-modules with $M''$ flat. ...
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12answers
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Proving that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic.

I am trying to prove that $\mathbb{Z}[x]$ and $\mathbb{Q}[x]$ are not isomorphic. I was thinking of arguing the following: Suppose there exists an isomorphism $\varphi: ...
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0answers
49 views

What is the proper name of this class of rings?

There is a property of rings that seems to be quite natural, but I can't seem to find a short name for it. A commutative ring with unit $R$ has this property if and only if it has no divisors of $0$, ...
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3answers
690 views

Are there interesting rings without unity?

There are several introductory textbooks which define a ring without any reference to a unity. However, nearly all of the rings one encounters in various branches of mathematics are endowed with a ...
2
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1answer
310 views

Localization of $K[x,y|x^2-y^3]$ and $K[x,y|xy]$ at $\langle x,y\rangle$ and $\{\text{non-zero-divisors}\}$ (exercise in SICA)

In Greuel & Pfister's A Singular Introduction to Commutative Algebra, p. 38, there is written: So we have rings $$\begin{array}{l l} R_1:= K[x,y|x^2\!-\!y^3], & R_4:= K[x,y|xy],\\ R_2:= ...
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0answers
59 views

Noetherian localizations and extra-condition implies Noetherian

I'm trying to solve this question but I'm stucked: If a ring $R$ satisfies the following two conditions: i) For every maximal ideal $M$ of $R$, if $S = R\setminus M$ then $S^{-1}R$ is ...
6
votes
1answer
121 views

Showing that $\mathbb{C}[x,y]^{\mu_n}$ and $\mathbb{C}[x,y,z]/(xy-z^n)$ are isomorphic as rings

The problem: Let $\mu_n$ act on $\mathbb{C}[u,v]$ with weights $(1,-1)$. I would like to show that the rings $\mathbb{C}[u,v]^{\mu_n}$ and $\mathbb{C}[x,y,z]/(xy-z^n)$ are isomorphic. Explanation of ...
5
votes
2answers
927 views

Examples of Infinite Boolean Rings

I'm attempting to list examples of infinite boolean rings and I need some clarification. Firstly, is it possible to take an infinite direct product of the integers mod $2$ to get a boolean ring? ...
3
votes
1answer
35 views

Characteristic collection of rings?

I have been trying to study ring theory to improve my algebra. One problem I have is that I have poor intuition about the general structure of rings that are not commutative. Could someone with a good ...
4
votes
1answer
96 views

Isomorphisms of rings and their generators

It's been a while since I touched abstract algebra, so please correct me if I'm wrong here: if I wanted to construct an isomorphism between the polynomial rings with integer coefficients ...
4
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2answers
991 views

Describe all ring homomorphisms of:

a) $\mathbb{Z}$ into $\mathbb{Z}$ b) $\mathbb{Z}$ into $\mathbb{Z} \times \mathbb{Z}$ c) $\mathbb{Z} \times \mathbb{Z}$ into $\mathbb{Z}$ d) How many homomorphisms are there of $\mathbb{Z} \times ...
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2answers
132 views

prove that $a_0$ is a unit and that $a_1 , a_2 , .. a_{n}$ are nilpotents in $R$ . [duplicate]

let R be a commutative ring . let $p(x) = a_n x^n + a_{n-1} x^{n-1} +...+a_1 x +a_0 $ $\in$ $R[x]$ prove that , $p(x)$ is a unit in R[x] iff a_0 is a unit and $a_1 , a_2 ,... , a_n $ are nilpotents ...
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1answer
54 views

Difficulty Understanding Primary Modules

I have read that any irreducible sub-module $I$ of a Noetherian module $M$ is primary. However if we let $M = \mathbb{Z}_8$ and $I = \mathbb{4Z}_8$ this isn't true, because $I$ is irreducible, and ...
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1answer
232 views

General proof that a product of nonzero homogeneous polynomials is nonzero (under certain conditions).

Background, Notation, Definitions: Given a set $X$, I define the set $M(X)$ of monomials with $X$-indeterminates to be the set of elements of $\omega^X$ having finite support. Given $m_0,m_1\in M(X)$, ...
2
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3answers
111 views

Noetherian ring question.

Let $R$ be a noetherian ring and $I$ a proper ideal of $R$. Show that $R/I$ is noetherian. Note: This was the last assignment (due date already passed [April 30] so I'm not directly asking for hw ...
1
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1answer
47 views

Field, Euclidean division question.

Let $K$ be a field and $f \in K[x]$. Show that if there is some $a \in K$ such that $f(a)=0$, then $x-a$ divides $f$. My friend told me to use Euclidean division by $x-a$. Also show that a ...
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2answers
120 views

For a nilpotent $r\in R$, show $r$ is in every prime ideal and that $1-sr$ is a unit for all $s\in R$.

Let $p \subset R$ be a prime ideal. Prove that for any nilpotent $r \in R$, it follows that $r \in p$. One of my classmates told me to use induction. Also, show that for all $s \in R$, $1-sr$ ...
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2answers
192 views

Idempotents in a Quotient Ring

Let $R=\mathbb{Z}_p[x]/(x^p-x)$. Show that $R$ has exactly $2^p$ elements satisfying $r^2=r$. I know that for $f,g\in\mathbb{Z}_p[x]$, we have $f-g\in(x^p-x)$ if and only if $f(a)=g(a)$ for all ...
4
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1answer
136 views

About Artinian Rings

I'm studing commutative algebra by the text of Atiyah and Macdonald, and a doubt come at me and I can not prove neither find a counterexample, the problem is: If a ring (commutative with identity) ...
5
votes
2answers
159 views

Algorithmic approach to enumerating ideals in $\Bbb Z[x]/(m, f(x))$

I'm studying for my algebra quals this fall and keep encountering problems like the following: List all the ideals of $\mathbb{Z}[x]/(16, x^3)$. or List all the primes of ...
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2answers
460 views

Is the inverse of a fractional ideal still fractional?

Let $R$ be a Dedekind domain, $K$ the field of fractions of $R$, $\mathfrak{m}$ be a fractional ideal of $R$, i.e. a non-zero finitely generated $R$-submodule of $K$. We define ...
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0answers
80 views

How to show that semiring of sets is a semiring?

It is known that ring of sets(measure theory) is actually a ring by taking set operations, intersection and symmetric difference, as multiplication and addition. How can we do the same thing to ...
0
votes
1answer
67 views

Ring Isomorphism Proof

Let $p$ be a prime with $p \equiv 1 (\mod 4 )$. I am trying to show that $\mathbb{Z}[X]/(X^2 + 1, p) \cong \mathbb{Z}_p \times \mathbb{Z}_p$ is a ring isomorphism. I am not really sure how to ...