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 (1)

2
votes
2answers
96 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 ...
0
votes
3answers
177 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 $1$. ...
2
votes
3answers
188 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 ...
1
vote
2answers
29 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 $a$, ...
0
votes
1answer
41 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= ...
6
votes
2answers
256 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 ...
3
votes
0answers
79 views

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 ...
2
votes
5answers
239 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 ...
2
votes
1answer
324 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 ...
4
votes
0answers
79 views

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 ...
0
votes
2answers
65 views

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. ...
2
votes
1answer
185 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 ...
3
votes
1answer
115 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 ...
1
vote
1answer
55 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 ...
2
votes
1answer
54 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 ...
1
vote
2answers
58 views

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.
5
votes
2answers
240 views

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.
0
votes
2answers
51 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 $\mathbb{C}[x,y]/(x,y)$ ...
1
vote
2answers
41 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 ...
0
votes
0answers
41 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 ...
0
votes
2answers
35 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 ...
3
votes
2answers
109 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 $e_1,e_2$ ...
1
vote
1answer
43 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 ...
1
vote
1answer
99 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 ...
-1
votes
2answers
83 views

{0} is unique maximal ideal when F is field [duplicate]

Let $R$ be a ring. Show that R is a field if and only if $\{ 0 \}$ is the unique maximal ideal of $R$. Thank you
6
votes
1answer
65 views

Distributive nearring

A nearring is a ring-like structure $R$ such that $(R, +, 0)$ is a group (possibly non-abelian) $(R, \cdot)$ is a semigroup $(x+y)\,z=xz+yz$ (right-distributivity) In other words, three conditions ...
2
votes
2answers
696 views

Examples of prime ideals that are not maximal

I would like to know of some examples of a prime ideal that is not maximal in some commutative ring with unity.
2
votes
2answers
550 views

Prove: The pre-image of an ideal is an ideal.

Let $\phi : R \to S$ be a homomorphism. If $N$ is an ideal of $S$, then $\phi ^{-1} (N)$ is an ideal of $R$.
0
votes
1answer
66 views

Is $f: \mathbb{Z}_{p^n} \rightarrow \mathbb{Z}_{p^n}$, $f(x) = x^p$ a homomorphism of rings

Define $f: \mathbb{Z}_{p^n} \rightarrow \mathbb{Z}_{p^n}$, where $p$ is prime, with $f(x) = x^p$. Is $f$ a homomorphism under addition? Thank you
2
votes
2answers
65 views

Constructing a certain bijection

I'm not sure at all which theorem(s) could be applied in order to get started on the following problem. Any suggestion is very much appreciated. Suppose $k$ is a field and $A$ is a finitely ...
1
vote
0answers
47 views

Integral Domain and PID Proof

Prove that, in a domain, $(a)=(b)$ iff $a = bu$ for some unit $u$. By $(a)=(b)$, it also means that $a\mid b$ and $b\mid a$ so we can write them as $a=bu$ and $b=av$ for some $u, v \in R$ where $u$ ...
2
votes
1answer
57 views

Divisibility and Principal Ideal Domain Proof

Let R be a ring. Show that a|b iff $b \in (a)$ iff $(b) \subseteq $ (a). I first just want to write out what I know about this statement: a|b means that a divides b or a is divisible by b and there ...
0
votes
1answer
42 views

Homomorphism between a ring which is a boolean algebra and one which is not.

I remember reading in a textbook that there can exist a homomorphism between a ring which is a boolean algebra and one which is not. Can anyone give me some example of this.
0
votes
3answers
344 views

Maximal Proper Ideal is a field proof

Show that a proper ideal M of a commutative ring R is maximal if and only if R/M is a field. What I know: Because M is a proper ideal $M \neq R$. The ideal M is maximal if it is a maximal element ...
0
votes
1answer
98 views

Show that the mod p map is a ring homomorphism

Let p be a prime and let (mod $p$)$ : Z[x] \mapsto Z_p[x]$ be the mod-p map which sends any polynomial... $f(x) = a_0 + a_1x + a_2x^2 + · · · + a_nx^n \in Z[x]$ ... to the polynomial... $f(x)$(mod ...
1
vote
1answer
54 views

Are there homogeneous elements with two distinct grades?

In a graded ring $B=\bigoplus_{d\ge 0} B_d$, the element $0$ is homogeneous with grade $d$ for every $d\ge 0$, in fact since every $B_d$ is an additive subgroup of $B$, then it must contain $0$. Can ...
4
votes
0answers
87 views

Extension of Euclidean Domain in which irreducibles have minimal norm

For the ring of polynomials $F[x]$ over a field $F$, there exists a larger ring $\bar{F}[x]$, the ring of polynomials over the closure of $F$, in which irreducibles are linear polynomials -- that is, ...
1
vote
2answers
134 views

Classification of Rings [duplicate]

I am trying to classify rings of order 10. I believe the only possible ring is $\mathbb{Z_{10}}$. Thus I am trying to find a map from my ring $R$ to $\mathbb{Z_{10}}$. The most obvious map $f: R ...
2
votes
1answer
98 views

Nilpotent elements in a commutative ring [duplicate]

Let $R$ be a commutative ring. Show that for any $a,b \in R$ nilpotent that $a+b$ is also nilpotent in $R$. We know $a^n = 0$ for some n and $b^m = 0 $ for some m, so consider $(a+b)^{m+n} = ...
1
vote
1answer
88 views

Is the completion of $(k[x,y]/f)_\mathfrak{m}$ isomorphic to $k[[x]][y]/f$?

Let $k$ be an algebraically closed field and let $f\in k[x,y]$ be an irreducible polynomial with no constant term that is not a polynomial in $x$ alone. Is it the case that the completion of the ...
1
vote
2answers
82 views

Identifying a quotient ring.

Consider the Quotient ring $\mathbb{Z}[x]/(x^2+3,5)$. Solution: I first tried to take care of $(5)$ in the above ring. Therefor we can consider $\mathbb{Z_5}[x]/(x^2+3)$. Now and interesting point ...
1
vote
1answer
91 views

Principal prime ideal is generated by irreducible element

$R$ is an integral domain, $x\in R$ and $(x)=I$ is a prime ideal. Prove that $x$ is an irreducible element of $R$. So I assume $ab\in I$, with $a, b \in R$. Since $I$ is a prime ideal, either $a$ ...
1
vote
1answer
80 views

Example of a simple ring with a nonsimple subring?

Does there exist a simple ring with a nonsimple subring?
7
votes
7answers
797 views

Can someone explain ideals to me?

I'm struggling with the idea of ideals (both the definitions and the common notation). I'm in a basic collegiate algebra course, just looking for a bit of help. As simply defined as possible, if you ...
17
votes
2answers
751 views

Proof that a certain subset of the reals is not a ring

Let $A = \{x \sin x : x \in \mathbb{Z}\} \subset \mathbb{R}$. Is $A$ a ring under the usual addition and multiplication operations of $\mathbb{R}$? It looks like it's not, but I can't find something ...
1
vote
1answer
54 views

Kernel and image of map on unit group of finite field

This question is from a rings and modules course I'm doing: Let $p>2$ be prime and $f:\mathbb{F}_{p}^{*}\rightarrow \mathbb{F}_{p}^{*}$ be given by $x \mapsto x^\frac{p-1}{2}$. Prove that $ker(f)$ ...
0
votes
1answer
58 views

Does the order of any element in a ring divide the order of 1?

I know that if a ring $R$ is a domain, then the additive order of any element must equal the additive order of $1$. If $R$ has zero divisors, then I can think of rings with elements whose additive ...
1
vote
2answers
86 views

co-idempotents: algebraic dual of an idempotent element?

So many times you can write out the axioms for an algebraic structure (say an algebra over a ring) as commutative diagrams and then reverse all the arrows and get a new structure: say a coalgebra. ...
0
votes
2answers
66 views

Quotient ring understanding

I just have a conceptual question regarding quotient rings and its elements. To get my point across, I will use the following example: Consider the quotient ring $\mathbb Z_5[x]/(x+1)^2$. Since the ...
3
votes
0answers
108 views

Counterexamples to the Artin-Rees Lemma

This well known Lemma about $I$-stable filtrations asserts: Lemma (Artin-Rees) Let $A$ be a Noetherian ring and $E$ a finitely generated $A$-module. Let $F$ be a submodule of $E$ and $\{E_i\}$ an ...