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

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Relation between finite stable rank and IBN (invariant basis number)

For R is any ring has an identity, we known if R has stable rank one, then R is "weakly finite" (or "stably finite," all matrix rings over R are Dedekind finite) and this implies R has IBN . But ...
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How to think of quotients of polynomial rings

I'm studying for an algebra midterm and I'm really just having a hard time wrapping my head around quotients of polynomial rings, especially ones where the ideal being quotiented by is something ...
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In a Commutative Ring, is Addition Necessarily Commutative?

In A First Course in Abstract Algebra, Fraleigh writes on p. 172 that "a ring in which multiplication is commutative is a commutative ring". Of course, this raises the question: is addition ...
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prime ideal as the intersection of ideals

I am studying for an upcoming exam, and I came across the following statement that I am struggling to prove: If $P$ is a prime ideal, then $P$ cannot be the intersection of two ideals that properly ...
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Prime Ideals and Maximal Ideals.

Let $n\in \mathbb{Z}$ be a non-zero integer. Let us define: $$\mathbb Z\left[\frac{1}{n}\right] := \left\{\left.\frac{a}{n^r}\, \right\rvert\, a,r\in \mathbb{Z}, r \geq 0\right\}.$$ What are ...
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Real analysis based on rings and ideals [duplicate]

Let $R$ be the ring of all the real valued continuous functions on the closed unit interval. Show that $ M=\{ f\in R:f(1/3)=0 \} $ is a maximal ideal
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Checking if $\langle 2 \rangle$ is a maximal ideal in $\mathbb{Z}[i]$

Is $\langle 2\rangle$ a maximal ideal in $\mathbb Z[i]$? Solution: We know that $\mathbb Z[i]$ is ED And hence PID. Consider $2\in\mathbb Z[i]$. Then $N(2)=2^2=4$ (NOTE: $N$ is norm). Since $N(2)$ is ...
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1answer
42 views

If M is free with a finite basis then every basis of M over R is finite and has the same number of elements.

Stuck on a proof in my lecture notes. Proposition: Let $R$ be a commutative ring and let $M$ be an $R$-module. If $M$ is free with a finite basis then every basis of $M$ over $R$ is finite and has ...
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1answer
87 views

Isomorphism of field of fractions

Assume that $1\in R$ I have a question which is as follows: Let $R$ be a commutative integral domain and $I$ a non-zero ideal of $R$. Let $F$ be the field of fractions of $R$. Suppose that ...
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2answers
57 views

Ideal in a ring of continuous functions

Let R be the ring of all continuous real valued functions on the unit interval $[0,1]$ (with pointwise operations), and let $I$ be a proper ideal of R. Show that there exists $λ \in [0,1]$ such that ...
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1answer
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If $S$ be a ring with identity and $f: M_{2}(\mathbb{R})\longrightarrow S$ is a nonzero ring homomorphism, which cases is true?

Let $S$ be a ring with identity and $f: M_{2}(\mathbb{R})\longrightarrow S$ is a nonzero ring homomorphism,then which cases is true? $S$ is left Artinian $S$ is left Noetherian $S$ is simple ring ...
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1answer
61 views

Units in a Subring of $\mathbb Q$.

I have this question: Let $n\in \mathbb{Z}$ be a non-zero integer. Let us define: $$\mathbb Z\left[\frac{1}{n}\right] := \left\{\left.\frac{a}{n^r}\, \right\rvert\, a,r\in \mathbb{Z}, r \geq ...
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2answers
35 views

I've found that R in my question is commutative but how can I check if R is a field?

Is R commutative? I answered yes because for both multiplication and addition R fulfills the commutative axiom. I didn't take the time to do every possible combination but I did write $0+0=0=0+0$ ...
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1answer
34 views

$IJ =(I\cap J)(I+J)$ holds in a PID? $I,J$ Ideals of a PID

One inequality is obvious, but the other one im not sure if holds. Any idea?
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2answers
14 views

Quotienting a direct sum by one of its factors

If we have a direct sum of $R$-modules. say $M_1\oplus M_2$ does it then follow that $(M_1\oplus M_2)/M_1\cong M_2$ This seems like it should be the case but I can't think of a way to prove ...
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1answer
45 views

Nilpotency of finite ideal

Suppose we have a commutative local ring $R$ with unit. I'm curious about whether the following statements are correct: 1- every proper finite ideal is nilpotent. 2-every proper finitely generated ...
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2answers
69 views

Non-Noetherian ring

I am wondering if there are applications of non-noetherian rings (domains). Are there areas where it is applicable ? Now obviously there is an academic interest in it. But I want to know if there are ...
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1answer
78 views

Describe all ring homomorphisms from Z x Z to Z x Z

Note: In this class, a ring homomorphism must map multiplicative and additive identities to multiplicative and additive identities. This is different from our textbook's requirement, and often means ...
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1answer
72 views

Ring homomorphism and ideals

Let $R$ and $R'$ be rings and let $\phi: R\mapsto R'$ be a ring homomorphism and $N$ an ideal of $R$. Show that $\phi[N]$ is an ideal of $\phi[R]$, and give an example to show that $\phi[N]$ need not ...
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46 views

Variety and algebraic curves

I am attempting the following problem from Artin: Every variety in $\mathbb{C^2}$ is the union of finitely many points and algebraic curves. I think the proof is trivial (unless I am missing ...
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1answer
67 views

Irreducible polynomials and algebraic geometry

I was reading Dummit and Foote and this was one of statements stated (without any proof), "An irreducible curve have finitely many singular points" I would like to know why is this true. Shouldn't it ...
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R is a commutative ring with unity and prime characteristic p, show that $\phi: R \to R\,\,/\,\, \phi(a) = a^p$ is a homomorphism

It's pretty obvious that $\phi(0) = 0$ and $\phi(1) = 1$ so those are all set. Now I want to show that $\phi(a+b) = \phi(a) + \phi(b)$ or $(a+b)^p = a^p + b^p$ for all $a,b \in R$. however it's ...
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1answer
100 views

Number of common zeros of two quadratic polynomials in ${\Bbb C}[t,x]$

The following theorem is in Artin's Algebra(2nd edition): Theorem 11.9.10 Two nonzero polynomials $f(t,x)$ and $g(t,x)$ in two variables have only finitely many common zeros in ${\Bbb C}^2$, ...
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integral basis for an arbitrary cubic Galois field

I wonder where I can find some information (possible a book) about finding an integral basis for cubic Galois fields? I know that for pure cubic fields there exists a simple criterion according to ...
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1answer
65 views

Problem about rings

Let $(R,+,*)$ be a ring so that $(x+y)^{2}=x^{2}+y^{2}$ $\forall\ x, y \in R$. Prove that A)$xy=-yx$ $\forall\ x, y \in R$ B)$x^{2}+x^{2}=0$ $\forall\ x, y \in R$ and $x+x=0\ \forall\ x, y \in R$ ...
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48 views

An integral domain $A$ which is also absolutely flat is a field

Question: Assume that $A$ is an integral domain such that every $A$-module is flat. Show that that $A$ is a field. Discussion: This seems to be very related to this question, in which it is shown ...
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147 views

Showing that the only idempotents in $R$ are zero and one

I have the following question that I have to solve however I cannot achieve. Let $R$ be a ring with $1$ and suppose $R$ has no zero divisors. Show that the only idempotents in $R$ are $0$ and $1$. ...
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3answers
98 views

How to prove idempotent element is nilpotent

I have a problem that I need to solve but I have trouble in solving the following question. Question is; Let $a \in R$ be a nonzero idempotent. Show that $a$ is nilpotent. ($R$ is a ring) I ...
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2answers
22 views

Domain with a minimal left ideal is a division ring.

I need to show that a domain $R$ with a minimal left ideal is a division ring. Suppose that $I$ is a minimal left ideal, then take $a\in I\setminus \{0\}$ and consider the left ideal generated by $a$, ...
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1answer
27 views

Greatest common divisors in Integral Domain

Let $R$ be an integral domain and $r,s\in R-\{0\}$ such that $\text{gcd}(r,s)=g.$ Suppose $\text{gcd}(kr,ks)$ exists, where $k \in R -\{0\}.$ Could anyone advise me on how to prove $kg= ...
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85 views

When is a local, reduced, (commutative) ring an integral domain?

Question I am wondering whether or not it is true that if $A$ is a reduced ring, then is it the case that the localization of $A$ at any of its prime ideals is an integral domain? Discussion ...
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A Commutator Identity in Rings

In a ring (or associative algebra), let the commutator $[A,B]$ be defined as $[A,B]=AB-BA$. I have asked earlier for a general formula for the expression $[x_1\cdots x_m,y_1\cdots y_m]$ in a group ...
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5answers
159 views

Maximal ideal contains a zero divisor

Suppose $R$ is a commutative and unital ring. Let the ideal $I$ be maximal and $a,b$ be (nonzero) zero divisors in $R$. Show that $ab = 0$ implies $a \in I$ or $b\in I$ We've ...
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1answer
158 views

Is any UFD also a PID?

Is there any counterexample that will disprove that every unique factorization domain (UFD) is also a principal ideal domain (PID)? I mean, any PID is a UFD, does the converse hold? Thanks in ...
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Chinese Remainder Theorem stuff (but more advanced, certainly derived from)

First, let $m$,$n$ be coprime. Suppose we want to find: $x\equiv y\text{ mod }m$ $x\equiv z\text{ mod }n$ As m,n are coprime $\exists a,b\in\mathbb{Z}:am+bn=1$ (GCD=1 basically) Then, for ...
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Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. [duplicate]

Let $f(x)$ belong to $\mathbb{Z}_p[x]$. Prove that if $f(b)=0$, then $f(b^p)=0$. Not sure how to proceed with this problem. I usually use Chegg, but Chegg doesn't have the solution for this problem. ...
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1answer
123 views

Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
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1answer
73 views

Showing $M\cong M'\oplus M''$ given an exact sequence

I am struggling with the following question: $R$ is a ring. $$M'\overset{f}{\longrightarrow} M\overset{g}{\longrightarrow} M''$$ are homomorphisms of $R$-modules such that for any $R$-module $N$, the ...
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1answer
37 views

Give an explicit ring isomorphism

I want to give an explicit isomorphism between $\mathbb{F}_7[X]/(X^2+2X+2)$ and $\mathbb{F}_7[X]/(X^2+X+3)$. I think the way to do it would be to send a root $\alpha$ of $X^2+2X+2$ to the ...
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1answer
53 views

May we write 4 when dealing with a ring that may not have a 4?

I have a question where I am asked to show that if for a ring $R$ and $\forall x\in R$ we have $x^2=x$ then $x+x=0$ I have shown this as follows: $(x+x)^2=x^2+x^2+x^2+x^2=x+x+x+x$ (by distributivity ...
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Is it possible that $R/I$ is a field when $R$ is non-commutative ring and $I$ is a maximal ideal of $R$?

Is it possible that $R/I$ is a field when $R$ is non-commutative ring with unit and $I$ is a maximal left ideal of $R$? If it is not, can anyone give an example of such $R$ and $I$? Thanks.
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Let $F$ be a field of order $2^n$. Prove that characteristic of $F$ is 2.

I figure that Lagrange's theorem and the fact that the characteristic of an integral domain is either $0$ or prime should be used, but just can't figure it out exactly.
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42 views

Quotient ring is infinite dimensional

I'm trying to show that $\mathbb{C}[x,y]$ is not a principal ideal domain. So I'm looking at the ideal $(x,y)$ and trying to show it is not principal. It is easy to see that $\mathbb{C}[x,y]/(x,y)$ ...
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31 views

A problem on $\text{UFD}$

Let $R$ be a $\text{UFD}$, and let $a,b,c \in R$ such that $1=\text{gcd}(a,b).$ Suppose $a |c, \ b|c.$ Could anyone advise me on how to prove $ab |c \ ?$ How do I use the fact that every nonzero non ...
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0answers
37 views

Nilpotent Subring [duplicate]

The question is : Show that the nilpotent elements of a commutative ring form a subring. Here is my unsuccessful take on it: Let $R$ be a commutative ring and let $S = \{a \in R | a^n = 0 \}$ be ...
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28 views

Common divisors in a PID

Suppose that R is an integral domain and that α, β, γ ∈ R. We say that γ is a common divisor of α and β if γ|α in R and γ|β in R. Suppose that R is a PID. Suppose that α, β ∈ R. Let I = (α) and ...
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81 views

Orthogonal idempotents from disjoint union in $\text{Spec}(A)$

Let $A$ be a commutative ring with unity. Suppose that $X=\text{Spec}(A)$ is a disjoint union $X_1\cup X_2$ of topological spaces. Show that $A$ has a pair of orthogonal idempotents $e_1,e_2$ ...
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1answer
38 views

Why does the only maximal of $k[[X_1,\ldots,X_n]]$ is $(X_1,\ldots,X_n)$?

I'm trying to understand in this book why the only maximal of $k[[X_1,\ldots,X_n]]$ ($k$ field) is $(X_1,\ldots,X_n)$: If I prove $rad(k[[X_1,\ldots,X_n]])\subset (X_1,\ldots,X_n)$, (where $rad$ is ...
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1answer
55 views

Strong Morita equivalence - Question about proof in Beer's “On Morita equivalence of nuclear $C^*$-algebras”

I'm going over the proof of this theorem about strong Morita equivalences on page 253 of "On Morita equivalence of nuclear $C^*$-algebras" by Walter Beer (http://bit.ly/1fOZiOw), I want to make sure I ...
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1answer
41 views

Please help me (proof or disproof these statement )

Please help me (proof or disproof these statement ) Let D be an integral domain , if there is an integer $n\in\mathbb{N}$ such that $na = 0$ for some $a\in D-\{0\}$, then characteristic of $D$ is ...