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
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Showing that $ab=0$ implies $ba=0$ in a ring

Question: Let $n$ be an integer greater than $1$. In a ring in which $x^{n}=x$ for all $x$, show that $ab=0$ implies $ba=0$. $(ab)^{n}={ab\cdot\cdot\cdot ab}=a(ba\cdot\cdot\cdot ba)b$ there ...
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2answers
22 views

The polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over the field $K$ has no zero divisors

Show that the polynomial ring $K[t_1,\dots,t_n]$ of $n$ variables over a field $K$ has no zero divisors (except the zero polynomial). When revising some Linear Algebra topics, I got stuck with this ...
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2answers
54 views

If every intermediate ring of a field extension is a field, then the extension is algebraic

Suppose $E/F$ is an extension of fields. Prove that if every ring $R$ with $F\subseteq R\subseteq E$ is a field, then $E/F$ is an algebraic extension. I can show the converse is true by ...
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0answers
40 views

Image of element not square of any element, maximal ideal, field is quadratic extension?

This is a followup to my question here. Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us ...
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0answers
27 views

How can we conclude from that that $I$ is a principal ideal? [duplicate]

Let $R$ be a commutative ring with unit. I want to show that if $I$ is an ideal of $R$ then $I$ is a free $R$-module iff it is a principal ideal that is generated by an element $a$ that is not a ...
3
votes
1answer
29 views

Rings and Rngs: properties which differ depending on the inclusion of a multiplicative identity.

As far as I know, one can define a ring with or without a multiplicative identity. My question is: what kind of properties, theorems, etc. get lost when one talks about rngs instead of rings, and ...
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1answer
22 views

If $R/I$ and $R/J$ are Noetherian (resp. Artinian), then so is $R/(I \cap J)$

Let $R$ be a commutative unitary ring and $I,J$ be two ideals of $R$. I need to prove that if $R/I$ and $R/J$ are Noetherian (resp. Artinian), then so is $R/(I \cap J)$ It seems that a direct ...
4
votes
1answer
36 views

Image of element is square of an element, precisely two maximal ideals satisfying condition.

Say we have $\mathbb{F}_q$ a finite field, $\text{char.} \neq 2$, have $f \in \mathbb{F}_q[x]$, $f \notin \mathbb{F}_q$ be a squarefree element, and let us look at the ring $\mathbb{F}_q[x, ...
2
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2answers
52 views

A ring isomorphic to a proper subring of itself

Give an example of a ring which is isomorphic to a proper sub-ring of itself. HINT: Consider $\Bbb R^\Bbb N$. My try:As given in the hint I considered $\Bbb R^\Bbb N$ i.e the set of all ...
3
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0answers
49 views

The GCD of a Univariate Integer-Valued Polynomial over a Set

Let $\mathcal{I}[X]$ denote the subring of $\mathbb{Q}[X]$ consisting of all integer-valued polynomials (i.e., $f(X)\in \mathbb{Q}[X]$ such that $f(k)\in\mathbb{Z}$ for all $k\in\mathbb{Z}$). For ...
1
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0answers
41 views

Generalized Jacobian Conjecture

Is there any known generalization of Jacobian conjecture which gives condition for $k[f_1, \ldots, f_m] = k[g_1, \ldots, g_m]$ where all $f_i$ and $g_i$ are functions over $x_1, \ldots, x_n$. Note ...
0
votes
1answer
21 views

local-global rings and its examples

i cant associate definitions of local-global rings,and also cant understand its exeamples.i am so confused,does anyone know about it? defn1: A commutative ring is local-global,if every polynomial ...
-1
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1answer
49 views

The endomorphism ring is a field [closed]

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that the endomorphism ring $\text{End}_R(M)=\text{Hom}_R(M,M)$ of a simple $R$-module is a field. $$$$ We have ...
1
vote
1answer
36 views

Show that it is equal to $\text{Im}f\oplus \ker g$

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that if $f:M\rightarrow N$ and $g:N\rightarrow M$ are $R$-module homomorphisms such that $gf=1_M$ then ...
0
votes
0answers
50 views

Show that it is equal to $\ker f \oplus \text{Im}f$ [duplicate]

Let $R$ be a commutative ring with unit and $M$ be a $R$-module. I want to show that if $f:M\rightarrow M$ is a $R$-module homomorphism such that $f^2=f$ then $M=\ker f \oplus \text{Im}f$. $$$$ ...
0
votes
1answer
24 views

Quotient ring example

Theorem: Let R be a ring and let I be an ideal of R. Define $\frac{R}{I}=\left \{ a+I \mid a \in R \right \}$ with binary operation : $\left ( a+I \right )+\left ( b+I \right )=\left ...
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0answers
13 views

Structure of ring [duplicate]

Determine the structure of the ring $R_0$ obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ satisfying each set of relations: $$ \begin{cases} 2x-6 &= 0\\ x-10 &=0 \end{cases} ...
0
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0answers
45 views

$F$ is a field, $n>1$ is it true that $\mathrm{Aut}(F[x]/(x^n))=F^{\times}$?

My question is to ask for whether the following generalization of this problem is true or not. The mentioned problem says $A:=\mathrm{Aut}(\mathbb{Q}[X]/(X^2)) \simeq \mathbb Q^{\times}$ by proving ...
0
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1answer
33 views

Annihilator - Product of cyclic groups

Let $M$ be the abelian group, i.e., a $\mathbb{Z}$-module, $M=\mathbb{Z}_{24}\times\mathbb{Z}_{15}\times\mathbb{Z}_{50}$. I want to find the annihilator $\text{Ann}(M)$ in $\mathbb{Z}$. $$$$ ...
0
votes
1answer
30 views

Find a polynomial $g(x) \in \Bbb Q [x]$ such that $I = g(x) $

Find a polynomial $g(x) \in \Bbb Q [x]$ such that ideal $I = (g(x)) $, where $I = \{f(x) \in \mathbb Q[x] : f(\sqrt2) = 0\}$ $ I = \{f(x) \in \mathbb Q[x] : f(1-i) = f(1+i) = 0 \}$ For 1, I think ...
2
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0answers
34 views

Generators of $ Z[x_1, x_2,\ldots , x_n] $

Is there any characterization of n-element generators of $ Z[x_1, x_2,\ldots , x_n] $? They obviously need to be algebraically independent.
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2answers
48 views

Show that $x^{n-1} =1$ for all non-zero element in a field

Question: Show that $x^{n-1} =1$ for all non-zero element in a field Let F be a finite field of order n. Show that $x^{n-1}=1$ for all non-zero $x \in F$. We have $\left | F \right |=n.$ ...
0
votes
1answer
21 views

example of a commutative ring without zero divisor that is not an integral domain

I'm not sure if I understand this question. An integral domain is a commutative ring (with unity) without zero-divisors. The question ask for an integral domain that is not an integral domain? Can ...
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0answers
19 views

zero-divisor and unit for a direct product

Describe all units and zero-divisor of the ring $\mathbb{Z}\times \mathbb{Q}.$ Note $\mathbb{Z}\times \mathbb{Q}=\left \{ \left ( x,\frac{p}{q} \right ) \mid x\in\mathbb{Z},\mathbb{P} \in \mathbb{Z}, ...
1
vote
1answer
21 views

Can a Nonzero Element in $\mathbb Z[\omega]$ be Divisible by Arbitrarily Large Powers of $1-\omega$.

Question. Let $p$ be a prime and $\omega$ be a primitive $p$-th root of unity. Let $a$ be a nonzero element of $\mathbb Z[\omega]$. Can it happen that for each $n\in \mathbb N$, $(1-\omega)^n$ ...
0
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1answer
21 views

generator is an ideal of a polynomial ring

I=$\left \{ p\left ( x \right ) \in R[x] \mid p\left ( x \right ) \text{has constant term 0}\right \}$ $I=\left \langle x \right \rangle$ is an ideal of R[x] I want to verify this claim but run ...
0
votes
1answer
49 views

Each cyclic $R$-module is isomorphic to an $R$-module of the form $R/J$ [closed]

Let $R$ be a commutative ring with unit. I want to show that each cyclic $R$-module is isomorphic to an $R$-module of the form $R/J$ where $J$ is an ideal of $R$. $$$$ Could you give me some ...
2
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0answers
40 views

Units in a group ring $\mathbb{Z}[G]$

This is a homework question: For a group G, let $g\in G$ have finite order, such that $\langle g\rangle$ is not a normal subgroup of $G$. Then $\mathbb Z[G]$ has a unit other than $\pm h$ with ...
1
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1answer
40 views

Review on polynomial quotient rings: $F[x]/(x^{4}-2x^{2}+1)$ & $\Bbb C[x,y]/(xy)$

I have searched problems about quotient rings on our site. I think I now have a certain understanding about problem like (1) Find all ideals of $F[x]/(x^{4}-2x^{2}+1)$ when $F=\Bbb C, \Bbb R.$ ...
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0answers
23 views

Is $\prod_{\sigma\in \Sigma(L)}\mathbb Z$ free of rank $[K:\mathbb Q]$?

Let $L/\mathbb K$ be a Galois extension of number fields and $\Sigma(L)$ the set of embeddings of $L$ into $\mathbb C$. Let $H_L= \prod_{\sigma\in \Sigma(L)}\mathbb Z$. Let $G$ act on $H_L$ in the ...
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1answer
18 views

Characteristic of an integral domain is 0 (or prime) [duplicate]

The characteristic of an integral domain $R$ is $0$ (or prime). My lecture has not yet covered infinite integral domain but I'll like to understand the proof. Basic fact: $R$ is an ...
2
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1answer
42 views

A subring of $\Bbb{Z}[X]/(X^2)$ that is a integral domain?

I need to find a subring of $\Bbb{Z}[X]/(X^2)$ that is an integral domain. My initial thought was the trivial subring consisting of only one element namely $0+(X^2)$ but my definition of integral ...
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0answers
30 views

How to prove this in ring theory?

Let $\Bbb{Z}_7$ be the ring of integers modulo $7$ and let $\Bbb{Z}[w]=\{a+bw:a,b \in \Bbb{Z}\}$ with $w^2+w+1=0$. Let $x=2-w$ show there exists $c \in \{0,1,2,...,6\}$ and $d\in \Bbb{Z}$ such that ...
0
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2answers
20 views

Let $S$ be a subring of a commutative Noetherian ring $R$. Then $R$ is finitely generated as an $S$-module? [closed]

Let $S$ be a subring of a commutative Noetherian ring $R$. Then how can I show that $R$ is finitely generated as an $S$-module?
1
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1answer
38 views

Isomorphism between two polynomial quotient fields

What, generally, is the strategy for proving if two polynomial quotient fields are isomorphic? Say from $\mathbb Z_{11}[x]/\langle x^2+1\rangle$ to $\mathbb Z_{11}[x]/\langle x^2+x+4\rangle$? My first ...
0
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1answer
23 views

What is a Coefficient ring in Ring theory?

In lecture, a (well, actually, two words) word "coefficient ring"-without having been defined-was used in the theorem Theorem: If the coefficient ring D is an integral domain then so is its ...
3
votes
3answers
44 views

Is regular selection from recurrence also recurrence?

Let $R$ be a ring, and $u=(u(0),u(1),u(2),...)$ be a sequence over $R$ ($u(i)\in R$). Let $m\ge1$, $c_0,...,c_{m-1}\in R$ be fixed elements, and the following law of recursion holds ...
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2answers
34 views

Cancellation law of ideals in a certain ring

Let $R$ be a integral domain satisfying the following property. For any non-zero ideal $A$ of $R$, there exist $a \in R\ (a \neq 0)$ and a non-zero ideal $B$ of $R$ such that $AB=(a)$. ...
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3answers
41 views

How can I show the uniqueness of homomorphism?

Let $R$ be a commutative ring and let $k(x)$ be a fixed polynomial in $R[x]$. Prove that there exists a unique homomorphism $\varphi:R[x]\rightarrow R[x]$ such that $\varphi(r)=r\;\mathrm{for\; ...
2
votes
2answers
56 views

Fraleigh's proof that $ M $ is a maximal ideal if and only if $ R/M $ is a field

I was reading Fraleigh's abstract algebra textbook and he gave a proof about the theorem that if $ R $ is a commutative ring with unity, then $ M $ is a maximal ideal if and only if $ R/M $ is a ...
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0answers
10 views

The reduced norm map $\operatorname{Nrd}: K_1(A)\to K^\times$

Let $K_1(A)$ be the Grothendieck $K_1$-group of the category of finitely generated projective $A$-modules where $A$ is a central simple $K$-algebra. I'd be grateful if someone could tell me if the ...
0
votes
1answer
17 views

What is the reduced norm map?

This is a basic question about the reduced norm homomorphism. Let $A$ be a central simple $K$-algebra and $P$ a f.g. projective $A$-module. I know that $\operatorname{End}_A(P)$ is also a central ...
2
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2answers
68 views

$R$ commutative ring with unity , does polynomials with unit leading coefficients of degre s from $0$ to $n$ generate all polynomials of deg $\le n$?

Let $R$ be a commutative ring with unity , consider the polynomial ring $R[x]$ , let $\mathcal P_n:=\{f \in R[x] : f=0$ or $\deg f \le n\}$ , so $\mathcal P_n$ is a finitely generated module over $R$ ...
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1answer
24 views

We assume that there exists a ring homomorphism $f:k[x,y]/(\phi(x,y))\to k[t]/(t^2)$ that satisfy given conditions.

Let $k$ be a field, $r \in k$, and $\phi(x,y)=\sum a_{ij}x^iy^j\in k[x,y]$. We assume that there exists a ring homomorphism $$f:k[x,y]/(\phi(x,y))\to k[t]/(t^2)$$ satisfying: ...
2
votes
2answers
29 views

To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F[x,y]$ by showing that $F[x,y]/\langle x-a , y-b\rangle$ is a field

Is there any way to show that for $a,b \in F$ , the ideal $\langle x-a , y-b\rangle$ is maximal in $ F[x,y]$ , by showing that the quotient $F[x,y]/\langle x-a , y-b\rangle$ is a field ? Is the ...
3
votes
1answer
52 views

$(x+a)^n=x^n+a$ in $\mathbb Z/n\mathbb Z[x]$ then n is prime.

Question If $n>1$ is an integer such that $(x+a)^n=x^n+a$ in $\mathbb Z/n\mathbb Z[x]$ for all $a \in \mathbb Z/n\mathbb Z$ then $n$ is prime. I want to show this with contradiction. But ...
1
vote
3answers
47 views

Is $M_a=\{f:f(a)=0\}$ a Principal ideal [duplicate]

Is $M_a=\{f\in \mathcal C[0,1]:f(a)=0\}$ where $\mathcal C[0,1]$ denotes the ring of continuous functions in $[0,1]$ a Principal ideal? My try: Let $M_a=\langle f_1\rangle$ .Consider $f(x)=\sqrt ...
2
votes
1answer
33 views

Minimum cardinality module for a fixed finite ring

Let $F$ be a finite field and $k$ be a positive integer. Let $M_k(F)$ denote the ring of $k\times k$ matrices. $M_k(F)$ is an $M_k(F)$-module with matrix multiplication, and $F^k$ is an ...
0
votes
1answer
61 views

Every finite field can be obtained by a quotient of the ring $\Bbb Z[x]$.

Every finite field can be obtained by a quotient of the ring $\Bbb Z[x]$. My try: Any finite field $F$ is of the order $p^n$ where $p$ is a prime and $n\in \Bbb N$ . If we want to make a field ...
0
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0answers
11 views

Equivalent definitions of primary ideals [duplicate]

In Atiyah-Macdonald, an ideal $I\in R$ is primary if $fg\in I$ implies that $f\in I$ or $g^n \in I$ for some $n$. The ideal $I$ is $P$-primary for a prime ideal $P$ if it is primary and $\sqrt{I}=P$. ...