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
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 ...
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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
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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
115 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
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1answer
266 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
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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
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1answer
90 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
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1answer
246 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
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1answer
347 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 !
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0answers
296 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
163 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
79 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}) ...
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0answers
101 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
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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 $\{f_1,\ldots,f_n\}$ ...
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3answers
132 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 ...
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1answer
230 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 ...
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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 ...
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1answer
302 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 ...
4
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2answers
386 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
106 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
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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
857 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
66 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
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1answer
104 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
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2answers
772 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?
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2answers
735 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
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2answers
135 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 $n$?
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1answer
221 views

Find the multiplicative inverse of x in a quotient ring.

This isn't exactly homework but I'm reviewing past papers (without solutions for an exam), and need some help with the following question: Find the multiplicative inverse of x in the quotient ring ...
4
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1answer
222 views

the field $\mathbb{Z}_3[i]$ is ring-isomorphic to the field $\mathbb{Z}_3[x]/(x^2 + 1)$

Let $\mathbb{Z}_3[i] =${$a+bi | a, b \in \mathbb{Z}_3$} . Show that the field $\mathbb{Z}_3[i]$ is ring-isomorphic to the field $\mathbb{Z}_3[x]/(x^2 + 1)$ how can I able to do this?can someone ...
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1answer
351 views

Why is the additive identity of a ring always a multiplicative absorbing element?

In problems concerned with finding the units in a ring, my textbook seems to always ignore the additive identity as a possibility. In combination with learning the definition of a field (a ring in ...
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1answer
62 views

$F[x]/\langle p(x)\rangle$ is a field $\iff F[x]/\langle p(x)\rangle$ is an integral domain

$$\color{red}{Is~my~interpretation~correct?}$$ Let $F$ be a field. I know that $p(x)\in F[x]$ is irreducible $\iff \langle p(x)\rangle$ is maximal i.e. $F[x]/\langle p(x)\rangle$ is a field $\iff ...
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1answer
83 views

Two problems in ideal and radical

Let $R$ be a commutative ring with multiplicative identity. Let $I$ be an ideal of $R$. Let $S=\{r \in R: r^n \in I\mbox{ for some natural number }n\}$. Show that $S$ is an ideal of $R$. Give an ...
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0answers
141 views

associated graded ring is the quotient of a free algebra by a homogeneous ideal

Let $A$ be a semilocal ring with Jacobson radical $m$ and let $I$ be an ideal of definition, i.e. an ideal such that $m^{\nu} \subset I \subset m$. Consider the associated graded ring of $A$, given by ...
3
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1answer
72 views

Ring structure on subsets of the natural numbers

Let $$\mathcal{N}=\{\{k_1,\ldots,k_s\}:\ s>0,\ \mbox{and the}\ k_i\ \mbox{are non-negative and pairwise different integers}\}\cup\{\emptyset\}.$$ Note that there is a bijection with the naturals, ...
1
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1answer
113 views

maximal ideal properly contains union of its square with the union of minimal prime ideals

One of the first theorems one encounters in the study of commutative algebra is that if $I$ is an ideal of a ring $A$ not contained in any of the prime ideals $P_1,\cdots,P_n$, then $I$ is not ...
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1answer
324 views

Why ring with only even numbers is not an integral domain?

Let $S$ be a set of all even integers. According to my text book, $(S,+,\cdot)$ is a ring which is not an integral domain. It is stated as a fact without an explanation and I fail to see the reason ...
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1answer
39 views

Let $I$ be an ideal in $R$. Show that $\mathrm{ann}_{R}(R/I)=I$.

I am trying to show that $\mathrm{ann}_{R}(R/I)=I$, but not sure whether I am "cheating". I can do the inclusion $I\subseteq \mathrm{ann}_{R}(R/I)$. What I am concerned about is ...
4
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1answer
115 views

A question on factorial rings

Is 31 irreducible in the ring $\mathbb{Z}\left[\sqrt{5}\right]=\left\{a+b\sqrt{5}:a,b\in\mathbb{Z}\right\}$ ? And is it prime in $\mathbb{Z}\left[\sqrt{5}\right]$?
3
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2answers
196 views

Endomorphisms of a semisimple module

Is there an easy way to see the following: Given a $k$-algebra $A$, with $k$ a field, and a finite dimensional semisimple $A$-module $M$. Look at the natural map $A \to \mathrm{End}_k(M)$ that sends ...
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votes
4answers
248 views

Why is $\{a + b\sqrt2 + c\sqrt3 : a\in\Bbb{Z}, b, c \in\Bbb{Q}\}$ not closed under multiplication?

The set $R = \{a + b\sqrt{2} + c\sqrt{3}: a \in \Bbb{Z}, c, b \in \Bbb{Q}\}$ is not closed on multiplication, my textbook states. Why is this? And related to that: why then is $S = \{a + b\sqrt{2} : ...
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2answers
132 views

Proving that tensor distributes over biproduct in an additive monoidal category

I'm trying to prove that the tensor product distributes over biproducts in an additive monoidal category; namely that given objects $A,B,C$, we have $A \otimes (B \oplus C) \cong (A \otimes B) \oplus ...
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1answer
102 views

Are these prime ideals?

Let $R=\mathbb Z[\sqrt{-5}]$. I want to show $P=3\,R+(1+\sqrt{-5})\,R$ and $Q= 3\,R+(1-\sqrt{-5})\,R$ are prime ideals of $R$.
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1answer
58 views

$R=\mathbb{Q}[x]/I$ where $I=\langle 1+x^2 \rangle$ let $y$ be the coset of $x$ in $R$ then

$R=\mathbb{Q}[x]/I$ where $I=\langle 1+x^2 \rangle$ let $y$ be the coset of $x$ in $R$ then $y^2+1$ is irreducible over $R$ $y^2-y+1$ is irreducible over $R$ $y^2+y+1$ is irreducible over $R$ ...
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1answer
97 views

Completion of rings

Theorem 10.1, p.52 of Lang's book on Algebra proves that $\hat{R}_I=\varprojlim_n R/I^n$ is ring ismorphic to Cauchy sequences modulo null sequences, he calls the latter the completion with respect to ...