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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Minimal number of generators and associated prime ideals

Let $R$ be a commutative Noetherian local ring, $M$ a finitely generated $R$-module with minimal number of generators $\nu_{R}(M)=s$, and let $Q$ be the total quotient ring of $R$. Then, ...
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1answer
26 views

Function from a set to a ring

Let $A$ be a ring and $S$ be a set. Let $F(S,A)$ be the set of functions of the form $f: S \rightarrow A$. It's defined that for $f,g \in F(S,A)$: $(f+g)(s)=f(s)+g(s)$, and, $(fg)(s)=f(s)g(s)$ Show ...
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principal ideals, integral domains, ideals,?

I am stuck trying to grasp this concept. I know that $\Bbb{Z}$ is a PID, $R=\Bbb{Z}[X]$ is not a PID, $\Bbb{Z}[i]$ is a PID. If someone could help me grasp these concepts it would be helpful. ...
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2answers
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Finding a ring isomorphism

Let $\phi : R \to R'$ be a ring epimorphism and $J\lhd R'$ an ideal of $R'$. Indicate a ring isomorphism $\psi: R/\phi^{-1}(J) \to R'/J$ The only thing i know about this problem is that ...
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1answer
46 views

Ideals on Rings?How do i Define them?

How do I define all the possible ideals of a given Ring-Set? Example on $Z(m)$. Do I stop when I find enough ideals that their union give's me my given set??
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3answers
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$x^p-x+a$ irreducible for nonzero $a\in K$ a field of characteristic $p$ prime

Is it true that $f(x)=x^p-x+a\in K[x]$ is irreducible for nonzero $a\in K$ a field of characteristic $p$ prime? I've seen variants of this question around, but they don't seem to answer the question ...
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Evaluation maps over polynomials

just looking for feedback and/or hints about this proof I've been working on. No answers please, but I'd like to know if I'm on the right track here. So I'm given a field $F$, and a non-zero $n ...
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3answers
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Polynomials in $\Bbb Q[x]$ with same real root dont have common divisor with degree more than 1

Let $f,g\in \Bbb Q[x]$ polynomyals with the same real root $\alpha \in \Bbb R$. I'm asked wether or not $f$ and $g$ must have a common divisor $h\in \Bbb Q[x]$ with $\deg(h) \geq 1$. I believe that ...
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1answer
14 views

About the connection between ideals and homomorphisms

I know that for a homomorphism of rings $\psi : R\rightarrow S$ we have that $\ker\psi$ is an ideal of $R$. I was wondering if the opposite direction is true: Let $I$ be an ideal of $R$. Then does ...
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2answers
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Find a ring homomorphism from $ \mathbb{Z}[x] \to \mathbb{Z}_7[\sqrt{3}] $

So far I'm thinking of using the evaluation at $ \sqrt{3} $ which is clearly a homomorphism to the ring $ \mathbb{Z}[\sqrt{3}] $ however I'm having trouble proving it to be a homomorphism to $ ...
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1answer
40 views

Ideals generated for commutative ring

The following is a problem from the Gallian book. I'm trying to understand what exactly this ideal is and how to verify that it is in fact an ideal. "Let $R$ be a commutative ring with unity and let ...
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1answer
25 views

Is there a direct proof that $M_n(\mathcal k)$ is semisimple ring.

An R-module M is called semisimple if on of the following condition holds: 1) M is a direct sum of simple* submodules of M 2) M is a sum of simple submodules of M 3) For any R-submdoule N of M ...
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1answer
118 views

Why this Tensor product of fields is a field

Is it true that the ring $\mathbb{Q}[i]\otimes_\mathbb{Q}\mathbb{Q}[i]$ is a field ?
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3answers
96 views

Quotient rings of polynomial rings

I have come across a quite difficult question while I am studying for a test: Let $F=\Bbb Z[x]/(7,x^2-3)$. Let $u$ denote the image of $x$ under the canonical epimorphism from $\Bbb Z[x]$ to ...
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4answers
86 views

$2 \mathbb{Z}$/$6 \mathbb{Z} $ is isomorphic to $\mathbb{Z_3}$

I am trying to understand how $2 \mathbb{Z}$/$6 \mathbb{Z} $ is isomorphic to $\mathbb{Z_3}$ . So far I understand that: $2 \mathbb{Z}$/$6 \mathbb{Z} $ = { $0+6\mathbb{Z}, 2+6\mathbb{Z}, ...
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1answer
15 views

Integral closure of a subring that is a polynomial ring over an algebraically closed field.

Let $K$ be an algebraically closed field that is a subring of an integral domain $D$. Assume $D$ contains an element $d$ that is transcendental over $K$. Also assume that $D$ is integral over ...
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1answer
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Exercise from Rotman: formal power series ring as inverse limit

Let $A$ be a commutative ring with unit, $J = (x)$ an ideal of $A[x]$. Thus we can consider the inverse system defined as $$\psi_{n,m}: A[x]/J^m \to A[x]/J^n$$ $$g(x) + J^m \to g(x) + J^n$$ $$\forall ...
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0answers
61 views

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$?

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$ ? $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)=\{ (0),\ (\tilde{x}-a,\tilde{y}-b),\ b^2=a^3\}$.
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1answer
29 views

Binomial Theorem on a Ring with Order 2

Say I have a Ring with set $G$ and binary operations $+$ and $\times$. If $G$ has order 2 under addition (meaning $A+A=0,\forall A\in G$, where $0$ is the additive identity), how can I reproduce the ...
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Rings having the same characters but not isomorphic.

I want to show that these two rings have the same characters but they are not isomorphic for $\nu>2$ Thank you for helping. $$H=k+kt^{4\nu}(1+t)+kt^{6\nu}(1+t)+kt^{7\nu}(1+t)+k[[t]]t^{8\nu}$$ ...
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2answers
67 views

Local ring of Krull dimension zero

In commutative rings text books it is usually asked to prove that as long as $(R,m)$ is a Noetherian local ring, the following are equivalent: (i) $m^n=m^{n+1}$ for some integer $n$; (ii) ...
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1answer
25 views

Are these properties for a monoid enough for being the underlying monoid of an integral domain minus the zero?

If $R$ is an integral domain then $R-\{0\}$ equipped with the original multiplication can be recognized as a commutative and cancellative monoid. The inversible elements form a subgroup $R^*$ and it ...
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1answer
44 views

Showing that two rings are not isomorphic

I have two rings: $R_1 = \mathbb{Z}_2[x]/\langle x^4+1\rangle$ and $R_2 = \mathbb{Z}_4[x]/\langle x^2+1\rangle$. I've shown that these have the same number of elements. Now I am struggling to show ...
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1answer
41 views

Do polynomials make sense over non-commutative rings?

One could think of polynomials rings as sort of a derived ring (a ring of functions $f: \mathbb{N}^m \mapsto R$ such that $f^{-1}(R \setminus \{ 0 \} )$ is finite), but from what I can tell, we are ...
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2answers
78 views

For a ring if we have: $\forall a\in R$ with $za=az=z$ does that mean $z=0$

For a ring if we have: $\forall a\in R$ with $za=az=z$ does that mean $z=0$? If we have $x+z=x$ for all $x\in R$ ($x+z=z+x$ as it is a ring) then we show z is unique and call it 0. For 0 it is ...
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1answer
409 views

How to prove that the inverse of a matrix is unique?

The ring of matrix is not an integral domain. How to prove that the inverse is unique?
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2answers
36 views

Sum of unit and nilpotent element in a noncommutative ring.

Something similar is asked here but it is not exactly the same thing. An element $a$ of a ring $R$ is called nilpotent if $a^n=0$ for an $n \in \mathbb{N}$. Show that if $u$ is a unit and $a$ is ...
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1answer
63 views

How to prove Ass$(R/Q)=\{P\}$ if and only if $Q$ is $P$-primary when $R$ is Noetherian?

Let $R$ be a Noetherian ring, $Q$ an ideal of $R$. How to prove that $$ \text{Ass}(R/Q)=\{P\} $$ if and only if $Q$ is $P$-primary? Update In fact, I have proved that if $Q$ is primary, then ...
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1answer
47 views

Symmetric powers of ideal quotients in a local ring.

Let $R$ be a local ring and $I \subset R$ any ideal. When is it the case that $(I \: \backslash I^2)^n = I^n \: \backslash I^{n+1}$? Put another way, when is the natural map $\text{Sym}^n(I/I^2) ...
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1answer
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History of the terms “prime” and “irreducible” in Ring Theory.

In ring theory, a nonzero, nonunit element $p$ of a integral domain is called irreducible if $p=ab$ implies that exactly one of $a$ and $b$ is a unit, and it's called prime if $p\mid ab$ implies that ...
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1answer
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Soft Question: Geometry of Rings

So.. I was thinking about something last night and literally have no idea where to start. Can the ring axioms be connected to topology? If so, how? Basically, the motivation for my question has to ...
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3answers
64 views

$A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal.

How to prove : $A/m^n$ is Artinian for all $n\geq 0$ if $A$ is a Noetherian ring and $m$ maximal ideal. Any suggestions ?
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2answers
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$1+\sqrt{2}$ is a unit in $\mathbb{Q}[\sqrt{2}]$. True or False

I believe that the statement is True, and this is my argument: Since there exists an element $((1-\sqrt{2})/(1-(\sqrt{2})^2)\in\mathbb{Q}[\sqrt{2}]$ such that their products gives $1$ (multiplicative ...
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1answer
55 views

Retraction for rings?

For abelian groups, the existence of left inverse or right inverse of a homomorphism can be characterized by looking at whether the image or kernel splits the group. Is there an analogous ...
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2answers
124 views

$\mathbb{Z} _{29}$ is a field. True or False.

My answer was True and this is my argument: Since $\mathbb{Z}_{n}$ has got $2$ operations plus the other properties of a ring, I figured that it is indeed a ring. On the other hand, since ...
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0answers
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Is $\min \deg$ in a dirichlet subring interesting or is it always $1$?

Let $s \in C$. Let $D = A[[n^{-X}]]$ be a subring of the formal (or absolutely converging on a region; whatever is needed) Dirichlet series with base ring $A$. Define a minimal Dirichlet series for ...
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1answer
27 views

Comparing injective dimensions in a short exact sequence

If $0→A→B→C→0$ is an exact sequence in the category of $R$-modules ($R$ commutative having unity) with injective dimensions of $A$ and $C$ both $≤n$, is that of $B$ also $≤n$? It seems to me that ...
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1answer
50 views

Give an example of a ring in which there exist $x,y,z \in R\setminus\{1,0\}$ such that $x$ is a unit, $y$ is nilpotent, and $z$ is idempotent.

Give an example of a ring in which there exist $x,y,z \in R\setminus\{1,0\}$ such that $x$ is a unit, $y$ is nilpotent, and $z$ is idempotent. I have an idea of the form $R = \mathbb{Z}/ ...
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1answer
77 views

Suppose a finite ring $R$. Show that each $x \in R$ is exactly one of a unit, nilpotent, or $x^k$ is idempotent

Suppose a finite ring $R$. Show that each $x \in R$ is exactly one of a unit, nilpotent, or $x^k$ is idempotent. I know I must show this in cases. Case 1: Suppose $x$ is a unit. Then there ...
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Matsumura, Exercise 18.8: Cohen-Macaulay and (not) Gorenstein

I need an answer to the exercise 18.8 of Matsumura's book: Let $k$ be a field and $t$ an indeterminate. Consider the subring $A = k[[t^3, t^5,t^7]]$ of $k[[t]]$ and show that $A$ is a ...
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1answer
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if $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about m zcn's comment on my question Projective dimension of all principal ideals is finite. Is R an integral domain?. It's a good point. so i ask it for use of everybody: if ...
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1answer
60 views

True or False? … [closed]

Let be $G$ a group, if $H$ is normal subgroup of $L$ and $L$ is normal subgroup of $G$, then $H$ is normal subgroup of $G$ Let be $f,g$ in $\mathbb Q[x]$, show that if $gcd(f,g)=d(x)$ then there ...
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0answers
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Readings for Noether

I'm studying the theory of Noether but I have only 4 pages of lecture notes with no details or examples. Are there any good lecture notes or chapters you know about? In my lectures the basics of ...
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1answer
117 views

How to calculate $\operatorname{Spec} \mathbb{C}[x,y]/(y^2-x^3)$

Is there a general method for calculating things like $\operatorname{Spec} \mathbb{C}[x,y]/I$ ? Maximal ideals are $ \{(x-\tilde{a},y-\tilde{b}): b^2-a^3=0\}$ because of ...
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2answers
50 views

Algebra - Gaussian integers

Let $\mathbb{Z}[i]=\{ a+bi : a,b \in \mathbb{Z}\}$ be the ring of Gaussian integers. Let $x,y \in \mathbb{Z}[i]$ with $y \neq 0$. Show that there exist $q,r \in \mathbb{Z}[i]$ such that $x = yq + r$ ...
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3answers
33 views

Each $c \in R^*$ divides each polynomial of $R[X]$

We consider that $R$ is a commutative ring with $1_R$. Each $c \in R^*$(if we see it as a constant polynomial), divides each polynomial of $R[X]$. ($c \in R^*$ means that $c$ is invertible.) I ...
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1answer
30 views

A cyclic right ideal which is not finitely generated

I am looking for a two-sided ideal $I$ in a ring with identity such $I$ is not finitely generated as a left ideal but it is cyclic as a right ideal.
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1answer
33 views

Principal ideals as generated groups

This seems like a pretty simplistic question, but I can't find a solid, non-ambiguous answer to it. The question I'm given: Is $I$ a principal ideal of $R$? Given: $R=\mathbb{Z}$ and $I=\left\langle ...
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0answers
25 views

Brauer Group - A measure of complexity?

I have seen many authors state that the Brauer Group in some way measures the complexity of a field. I've convinced myself that the Brauer group of the reals is Z/2Z, and that the Brauer group of an ...
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1answer
55 views

A statement equivalent to flatness

If $R$ is a ring with identity and $P$ is a flat right $R$-module, it is a fact that any $R$-homomorphism $f$ from a finitely presented right $R$-module $M$ to $P$ factors through a finitely generated ...