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
264 views

Irreducible elements are not associates

I would like to know if every irreducible elements in a ring are not associates. I'm asking that because of this part in the page 61 of this book: Thanks a lot
4
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1answer
129 views

Prove: If $x$ is not a left zero divisor and $xy=1$, then $yx=1$.

Let $R$ be a ring with unity, and let $x,y \in R$ with $xy = 1$. Assume that $x$ is not a left zero-divisor. Prove that $x$ is a unit. $$xy = 1$$ $$\begin{gather} xyx = x \\ xyx - x = 0 \\ x(yx-1) = ...
3
votes
1answer
204 views

Prove that $R$ has unity [duplicate]

Let $R$ be a finite ring such that $x^2=x$ for all $x$ in $R$. Prove that $R$ has unity. I was able to show that it was commutative. Proof: $x^2=x$ $x^2-x = 0$ $x (x-1) =0 $ thus $x = 0$ or ...
3
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5answers
310 views

$I=(3,x^2+1)$ be an ideal of $\mathbb{Z}[x]$. Then $I$ is a proper ideal of $\mathbb{Z}[x]$

This is from an old exam. Here is the problem. Let $I=(3,x^2+1)$ be an ideal of $\mathbb{Z}[x]$. Then $I$ is a proper ideal of $Z[x]$ How does one go about showing that this is a proper ideal. ...
4
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0answers
185 views

Generation of left ideals in group rings

This looks like an elementary exercise on group rings (I heard it somewhere), nonetheless it seems to be non-trivial to me. Any references much appreciated. Suppose that we are given an (infinite) ...
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0answers
36 views

Can we always solve these linear algebra equations given rows of multiples?

We can find $n$ elements of multiplicative order $(n+1)$ modulo some large prime $p$, according to this question. Now I'm wondering if we can always perform linear algebra on the elements, as ...
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0answers
82 views

Image of a Euclidean domain

Considering Euclidean domains using a function $\varphi :E\setminus\lbrace 0\rbrace \rightarrow \mathbb{N}$ such that (i) $\varphi(a)\leq \varphi(ab)$ with $b\neq 0$. (ii) for all $a,b\in E$, $b\neq ...
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1answer
115 views

An example of Von Neumann regular ring with nilpotent element in noncommuative ring.

Give an example of Von Neumann regular ring with nilpotent element.
11
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2answers
455 views

A ring with no non-zero nilpotents and $(ab)^2=a^2b^2$ for all $a,b$ must be commutative

Given that $R$ is a ring with no non-zero nilpotent elements and has $(ab)^2=a^2b^2$ for all $a,b\in R$, show that $R$ is commutative. I have previously shown that, if $R$ is unital and has ...
3
votes
2answers
289 views

If $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$

Show that if $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$. Can I get some help to solve this problem
3
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1answer
513 views

homomorphic image of prime and maximal ideals

Is the homomorphic image of a prime ideal always prime? Is the homomorphic image of a maximal ideal always maximal? Can someone help me please to check these questions please. thank you for ...
6
votes
3answers
270 views

Which is easier to work out: determinant or inverse?

Suppose $A\in M_n(R)$ be a $n\times n$ matrix over some ring $R$. Which of the following two tasks is easier? to work out $\det(A)$; to work out $A^{-1}$. More specifically, I want to know the ...
3
votes
2answers
155 views

$\mathbb{Q}[x,y]/(y)$ is isomorphic to $\mathbb{Q}[x]$

How can I show that $\mathbb{Q}[x,y]/(y)$ is isomorphic to $\mathbb{Q}[x]$ as rings? I know that first isomorphism theorem is required here but I could not find the proper homomorphism.
14
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1answer
509 views

When is there a ring structure on an abelian group $(A,+)$?

Given an abelian group $(A,+)$, what are conditions on $A$ that ensure there is or isn't a unitary ring structure $(A,+,*)$? That is, an associative bilinear operation $* : A^2 \to A$ with an ...
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1answer
116 views

Show that if an ideal is free as a module then it is principal.

Here $\mathfrak a$ is an ideal of a commutative ring $A$. Show that $\mathfrak a$ is principal if it is free as an $A$-module.
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1answer
294 views

What is the order of quotient rings?

I have a question related to determine the order of a quotient ring in the following: Consider the quadratic ring $\mathbb{Z}[\sqrt{-n}]$, and $\alpha$ and $\alpha'$ are 2 irreducible ...
3
votes
2answers
118 views

Are Mod of Rings are the same to Quotient Groups?

I found they sometimes are both written in $A/N$ pronounced $(A \bmod N)$ A remainder ring? a quotient group? what's the difference? Are there any conventional problems in notations?
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1answer
81 views

Is $R \setminus P$ a multiplicative subset?

Let $S$ be a subset of the ring $R$; we say that $S$ is multiplicative if   (a) $0 \notin S$,   (b) $1 \in S$, and   (c) whenever $a,b\in S$, we have $ab \in ...
2
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0answers
84 views

A question about the consequence of Prime Avoidance.

I have found the following statement: Let $R$ be a Noetherian ring and $x$ is a non-zero divisor of $R$. Let $P$ be a prime ideal associated to $xR$. Then by Prime Avoidance there exists a non-zero ...
0
votes
1answer
43 views

subalgebras and finitely generated modules

Let $A$ be a $k$-algebra, $B$ be a subalgebra of $A$, and $K$ be a left ideal of $B$ which is finitely generated as a $B$- module. Is $AK$ necessarily a finitely generated $A$-module?
3
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1answer
113 views

Does this morphism necessarily give rise to a finite extension of residue fields?

Let $f:X\rightarrow Y$ be a morphism of finite type of locally Notherian schemes. Let $x\in X$ and $y=f(x)$. Recall that $f$ is said to be unramified if the map of stalks $g:\mathcal O_{Y,y} ...
2
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0answers
72 views

A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
2
votes
1answer
264 views

which of the following rings is a PID?

which of the following rings is a PID? $1$. $\mathbb{Q}[x,y]/(x)$. $2$. $\mathbb{Z} \times \mathbb{Z} $ $3$.$\mathbb{Z}[x]$ $4$. $M_2(\mathbb{Z})$,the ring of $2 \times 2$ matrices with entries in ...
8
votes
1answer
68 views

Can non-isomorphic abelian groups have isomorphic endomorphism rings?

I am aware that distinct Banach spaces $X$, $Y$, give rise to distinct operator algebras $B(X)$, $B(Y)$, but the proof seems to rely heavily on the use of projections and the Hahn-Banach theorem. So ...
4
votes
1answer
88 views

Flat closed immersion into a Noetherian scheme is open

Let $X$ be an irreducible Noetherian scheme. Consider some flat closed immersion into it. I want to show that it is also open, so that the morphism is surjective. I have a few thoughts, but I can't ...
6
votes
1answer
239 views

Finite set of zero-divisors implies finite ring

Show that any commutative ring $R$ having only $n$ non-zero zero divisors ($n\geq 1$) is finite and doesn't contains more than $(n+1)^2$ elements.
2
votes
1answer
342 views

Application of the Chinese Remainder Theorem

Three brothers A, B and C live together and they all love eating pizza. A has the habit of eating a pizza every 5 days, B every 7 days and C every 11 days. A and C both eat pizzas together on 3 ...
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2answers
77 views

If R is a PID and I is an ideal of R, then every ideal of the quotient ring R/I is a principal ideal.

Solution,ideas and hint would be greatly appreciated. Thanks !
4
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0answers
292 views

Regular Noetherian local rings are integral domains - questions about the proof

I am reading a proof that if $(A,\mathfrak m)$ is a regular local ring, then $A$ is an integral domain. I put the major questions I'm worried about in bold, but there are a lot of little things I'm ...
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1answer
105 views

Question on normal Noetherian local rings

Consider a normal Noetherian local ring $(A,\mathfrak m)$ of dimension $1$. I am working through a proof that such a ring is a principal ideal domain. Consider $x\in \mathfrak m \backslash \mathfrak ...
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1answer
68 views

Normal at every localization implies normal

I'm having some trouble with basic ring theory. Let $A$ be an integral domain and $\alpha$ an element of its fraction field integral over $A$. I am trying to understand a proof that $\alpha\in A$ ...
1
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1answer
158 views

Integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$

I am trying to compute the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i].$ I have managed to show that $\mathbb{Z}[i]$ is inside the integral closure, and I suspect it is the entire thing. Can ...
3
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0answers
78 views

Show that if $M$ is Noetherian then there $ n_{0} \in \mathbb{N}$ such that $n \geq n_{0}$, $0= \operatorname{Im}(f^{n}) \cap \ker(f^{n})$

Sean $M$ an $R$-module and $f: M \longrightarrow M$ an endomorphism of $M$. Show that if $M$ is Noetherian then there $ n_{0} \in \mathbb{N}$ such that $n \geq n_{0}$, $0= \operatorname{Im}(f^{n}) ...
3
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0answers
99 views

When powers of matrices are represented as a sum of integral matrices

There is given a ring $R$ and a subring $K$ with unit. We have a matrix $A$ of size $n$ over $R$. The characteristic of $R$ is $0$ or more than $n$. The statement is: If $A^m$ for any ...
2
votes
2answers
127 views

Sequence of irreducible polynomials in $K[X_{1},…,X_{n}]$ generates a prime ideal?

I was thinking about how a chain of irreducible polynomials in $K[X_1,\ldots,X_n]$, where $K$ is a field, behave with respect of being prime. What I mean is the folowing: If $\{f_1,\ldots,f_n\}$ ...
1
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3answers
131 views

Ring and Subring with different Identities [duplicate]

Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know ...
8
votes
1answer
228 views

Construction(s) of new integral domains from “old ones”

Given an integral domain $D$, there are several ways how to construct a new integral domain related to D. For example, one can consider a ring of polynomials/formal power series/formal Laurent series ...
1
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2answers
71 views

The action of a Galois group on a prime ideal of a Dedekind domain

This is a slight variant of a question I asked earlier. Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let ...
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0answers
82 views

The action of a Galois group on a prime ideal in a Dedekind domain

Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $B$ be the integral closure of $A$ in $L$. If ...
1
vote
1answer
286 views

Graded commutative $R$-algebras

Let $R$ be a commutative ring and $T$ a graded commutative $R$-algebra. This means that $\,T$ consists of a collection $\{T_n\}_{n\geq 0}$ of $\,R$-algebras, where the elements of $R_n$ are called ...
3
votes
2answers
377 views

On the direct sum of rings

Let $A,B$ be rings. Suppose that $$A\cong A\oplus B$$ Can I conclude that $B=0$, the trivial ring? If so, how can be proved? Thanks
4
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1answer
105 views

Is a semisimple A-module semisimple over its endomorphism ring?

Let $A$ be a ring, $M$ be a semisimple $A$-module and let $B=End_A(M)$. Show that $M$ is semisimple as a $B$-module. My thoughts so far are: if I can show that $B$ is a semisimple ring, then it ...
2
votes
1answer
138 views

Submodules of $\operatorname{Hom}_R(M,N)$ with $R$ a commutative ring.

Is there a way to characterize the submodules of the $\operatorname{Hom}_R(M,N)$? $M,N$ are arbitrary $R$-modules and $R$ a commutative ring, to assure that $\operatorname{Hom}$ will be an ...
11
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1answer
838 views

Every maximal ideal is principal. Is $R$ principal?

Let $R$ be a commutative ring with 1. If every maximal ideal of $R$ is principal, is $R$ a principal ideal ring?
5
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1answer
96 views

Question about the Euclidean ring definition [duplicate]

I recently came across the following definition for a euclidean ring: There exists a function $g:R\to\Bbb N_0$ with the following properties: 1.) $\forall x,y \in R$ with $ y \neq 0$ there exist ...
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0answers
65 views

Is an invertible ideal in a semi-quasilocal ring a principal ideal?

Let $R$ be a semi-quasilocal ring and $I$ be an invertible ideal of $R$. Is $I$ a principal ideal of $R$?
3
votes
1answer
102 views

Given a balanced bimodule $_SP_R$, is $R$ isomorphic to $\text{Hom}_S(P,P)$?

For clarity's sake, let me recall some definitions: Given two rings $R$ and $S$, we call a bimodule $_SP_R$ balanced if the ring homomorphisms $$\lambda_P:S \rightarrow \text{End}(P_R)$$ $$\rho_P:R ...
6
votes
2answers
755 views

Principal ideal and free module

Let $R$ be a commutative ring and $I$ be an ideal of $R$. Is it true that $I$ is a principal ideal if and only if $I$ is a free $R$-module?
6
votes
2answers
718 views

Show that a ring with disconnected spectrum is a product of two subrings. [duplicate]

It's an exercise from the book introduction to commutative algebra by Atiyah and Macdonald. If $\operatorname{Spec}(A)$ is disconnected, I'm asked to show that $A$ is a product of two subrings. I ...
6
votes
2answers
131 views

Symmetric and exterior powers of a projective (flat) module are projective (flat)

Assume that $R$ is a commutative ring with unity and $P$ a projective (flat) $R$-module. Why $\mathrm{Sym}^n(P)$ and $\Lambda^n(P)$ are projective (flat) for every $n$?