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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Is every left maximal ideal the annihilator of a simple left module?

In my version of Noncommutative Algebra, by Benson Farb & R. Keith Dennis, in chapter I, section 2 on the Jacobson radical, it is claimed that … each maximal left ideal $I$ is the annihilator ...
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54 views

Definition of a field homomorphism

Given a field $F$ of characteristic zero, say $F=\mathbb{R}$, what is the minimal requirement for a function $\mu:F\to F$ to be a field homomorphism? (Do we need to require two axioms, one for ...
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48 views

Annihilator and Projective Dimension

I was reading the book A Course in Ring Theory by Passman and in it is the following lemma; and after this lemma there's a example which I don't quite understand; The main thing that I don't ...
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55 views

Is this set a subring of $\mathbb{Z}\times\mathbb{Z}$?

Is the set $S = \{(x,-x) : x \text{ is an integer}\}$ a subring of $\mathbb{Z}\times\mathbb{Z}$? I am not sure where to start here. Is $\mathbb{Z}\times\mathbb{Z}$ a matrix? It doesn't seem ...
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25 views

Projective Dimension and Supremum

Here is a lemma that appears in A Course in Ring Theory by Passman. In the last section of the proof the writer shows that, $\mbox{pd }A_i\leq n\iff \mbox{pd }A\leq n$ and finishes the proof. I don't ...
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116 views

Rings (integral domain and fields)

True or false: (1) Every integral domain is a field (2) every field is an integral domain (3) the ring $\mathbb Z$ is a field. (4) the ring $\mathbb Z/(17)$ is a field. (5)The set $\{[0], [2], ...
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96 views

Is always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$?

Let $A$ be a commutative ring. Let $f \in A$. Let $A_f= A\left [ \frac{1}{f}\right ]$. Let $\hat{A}$ the $f$-adic completion of $A$. Is it always (even when $A$ is not noetherian) true that $$\hat{A} ...
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106 views

Let $R$ be a Noetherian ring. Then all finitely generated $R$-modules are Noetherian

Here is an excerpt of my lecture notes: " Claim I: Let $M$ be $R$- module and $N$ be submodule of $M.$ Then $M$ is Noetherian iff $N, \ M/N$ are Noetherian. Def: The ring $R$ is Noetherian iff the ...
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230 views

In a noetherian integral domain every non invertible element is a product of ireducible elements

I want to prove that in a noetherian ring $R$ which is also an integral domain, every non invertible element can be expressed as product of ireducible elements. I really do not know where to ...
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194 views

Can there be more ideals than elements of a ring?

Can there be more ideals than elements of a ring? This is related to my other question Having elements as Ideals . At first glance, it seems obvious that there would be less ideals than elements of a ...
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2answers
104 views

Isomorphism between Rings $\mathbb{Z}[\frac{u}{v}]$ and $\mathbb{Z}[\frac{1}{v}]$, u,v relatively prime

Let $u$ and $v$ be relatively prime integers, and let $R'$ be the ring obtained from $\mathbb{Z}$ by adjoining an element $\alpha$ with the relation $v\alpha=u$. Prove that $R'$ is isomorphic to ...
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1answer
70 views

Artin 2nd Ed. Problem 12.5.3

The problem says "Find the generator for the ideal of $\mathbb{Z}[i]$ generated by $3 + 4i$ and $4 + 7i$." I don't understand the question. It asks us to find the generator of the ideal, but then it ...
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1answer
32 views

Why is $J/J^{2}$ is a vector space over $R/J$

Let $R$ be a commutative local ring with $1$ and let $J$ be the Jacobson radical. I have it written in my notes that $J/J^{2}$ is a vector space over the field $R/J$ that is an $R/J$ module. Why is ...
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1answer
40 views

Are these polynomials irreducible over $\Bbb Z$?

I have received these problems and I'm not sure where to start: Are these polynomials irreducible over $\Bbb Z[X]$ ? a) $X^{65536} + 1$ (which is really (c) for n = 16) b) $X^{10} + X^9 + X^8 + ...
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1answer
40 views

Proving $(φ(x)\cdot ψ(x)) \cdot ω(x)=φ(x) \cdot (ψ(x)\cdot ω(x))$ where $φ,ψ,ω$ are polynomials on a ring $R[X]$

If I take $3$ random polynomials $φ,ψ,ω$ on a ring $R[x]$, I'm trying to prove associativity which is very obvious. But I have trouble on the algebra part with the sums. I know that given $2$ ...
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1answer
35 views

A problem on $\text{ACCP}$

Let $R$ be a commutative ring. Could anyone advise me on how to prove $R$ has $\text{ACCP}$ (Ascending chain condition for principal ideals) iff every collection of principal ideals of $R$ has maximal ...
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36 views

Proving $C$ (the center of $R$) is a subring of $R$

For the three axioms Is $0$ contained in C? I got that by putting $a=0$ $(0)(r)=(r)(0) = 0$ For is $a-b$ contained in $C$ and Is $(a)(b)$ contained in $C$ I' ve been playing around with the ...
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1answer
28 views

If $R$ is $\text{UFD},$ then $R[X,Y]$ is $\text{UFD}.$

Let $R$ be commutative ring with $1.$ Suppose $R$ is $\text{UFD}.$ Could anyone advise me on how to prove $R[X,Y]$ is $\text{UFD}\ ?$ Thank you.
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142 views

Properties of GCD in rings

Let $R$ be subring of integral domain $S.$ Suppose $R$ is $\text{PID}.$ Let $a\in R$ be a greatest common divisor of $r_1,r_2$ in $R$. ($r_1,r_2 \in R$, not both zero). Could anyone advise me on how ...
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75 views

algebraic integer $\alpha$ + polynomial relation $\beta$ and $\alpha$ $\Rightarrow$ $\beta$ algebraic integer.

Assume $\beta$ can be expressed in terms of polynomial relation in $\mathbb{Z}[\alpha]$. Where $\alpha$ is an algebraic integer (i.e. $\alpha$ is the root of a polynomial in $\mathbb{Z}[X]$. How can ...
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0answers
49 views

some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
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2answers
81 views

If $M_1\cap K=M_2\cap K$ and $M_1+K=M_2+K$ then $M_1=M_2$?

I have a ring $R$ with $K\le M$ and submodules $M_1,M_2$. If we have that: $$M_1\cap K=M_2\cap K \text{ and } M_1+K=M_2+K$$ can we conclude that $M_1=M_2$? I don't think that this is true ...
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120 views

Is there a ring so that any two distinct non-zero elements do not commute?

I was wondering if there was a ring so that any two distinct non-zero elements do not commute. Formally, is there a ring $R\not=\{0\}$ so that $$\forall x,y\in R\setminus\{0\}, x\not= y\implies ...
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63 views

The ring of homogeneous polynomials

I think I found an error in my textbook, but I am not completely sure. The book is Hulek, Elementary algebraic geometry, pag. 73. There is a theorem showing that $U_i$ and the affine space ...
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1answer
128 views

Maximal ideals in the ring of eventually constant sequences of real numbers

For homework I am studying the ring $R$ of eventually constant sequences of real numbers (with multiplication and addition defined componentwise). What are the maximal ideals of $R$? By looking at ...
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2answers
37 views

Associates in Domains

Let D be a domain and $a, b \in D^*$. Show that $a$ is a proper divisor of $b$ if and only if $b=ax$ for some nonzero nonunit $x$. I'm just really not sure how to start this. Any advice would be ...
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70 views

Noetherian Ring and Homomorphic Image

Prove that, if $R$ is Noetherian, then so is each homomorphic image of $R$. I know that by the Fundamental Homomorphism Theorem this is the same as showing that if $R$ is Noetherian, then so is ...
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1answer
69 views

Polynomials over a field with characteristic $0$ is square free implies it's coprime with its formal derivative

Let $F$ be a field with characteristic $0$, $f \in F[t]$ the polynomial ring over $F$. Show that $f$ is square free implies $ f, f'$ are relatively prime. I know this is actually an if and only if ...
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89 views

A Lemma of Kaplansky

Source: Rings With a Polynomial Identity, Irving Kaplansky The Lemma: Suppose that $\mathbb{A}$ is an $\mathbb{F}$-algebra, where $\mathbb{F}$ is a field. Then, suppose that $\mathbb{A}$ ...
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120 views

How can I complete the proof of Noetherianity of I. S. Cohen?

Theorem (I. S. Cohen). If $R$ is an unital commutative ring, and for each ideal prime $\mathfrak{p}\in Spec(R)$ we know $\mathfrak{p}$ is finitely-generated as $R$-mod then $R$ is Noetherian. ...
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85 views

Regarding Irreducibility of two variable polynomial

Following is an example taken from Dummit Foote - Abstract Algebra after Proposition $9.4.12$ The idea of reducing modulo an ideal to determine irreducibility can be used also in several ...
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1answer
83 views

$f(x) \mid g(x) \iff g(x) \in \langle f(x) \rangle$. Isn't this trivial?

Let $F$ be a field and $f(x), g(x) \in F[x]$. Show that $f(x)$ divides $g(x)$ if and only if $g(x) \in \langle f(x) \rangle$. This seems... almost trivial to me (which is usually a sign that I'm ...
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49 views

Intersection of Ideals in a ring

It is clear that $I = \langle a \rangle \cap \langle b \rangle $ is an ideal. I am trying to prove that $a_1b_1 - a_2b_2 \in I \ \forall \ a_1b_1,a_2b_2 \in I $ It seems so easy but I am stuck. ...
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1answer
468 views

Polynomial rings over a field and maximal/prime ideals

Let $F$ be a field , I want to prove that every proper nontrivial prime ideal of $F[x]$ is maximal. My definitions of prime/maximal ideals are as follows: $N$ is a prime ideal of $R$ iff $ab \in N ...
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1answer
26 views

Degree 3 polynomial with coef in a field K: question on roots in a algebraic closure

I am asked the following question: Consider a field $K$ with characteristic different from 2 and 3, and the polynomial $f(t) = t^3 + pt + q \in K(t)$ with three distinct roots $\alpha_1, \alpha_2, ...
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1answer
110 views

A non-zero and non-invertible element in a noetherian integral domain has a decomposition into irreducible elements

Let $R$ be a noetherian integral domain. I want to show that any non-zero and non-invertible element $a$ can be written as a finite product of irreducible elements. my ideas: I should argue by ...
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1answer
713 views

Proving that the unities of a ring form a group under multiplication

I am presented with the following task: Show that if $U$ is the collection of all units in a ring $\langle R, +, \cdot\rangle$ with unity, then $\langle U, \cdot\rangle$ is a group. I am still ...
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36 views

Ring Embeds in Monoid Ring

Let $(S,+)$ be a nontrivial commutative monoid and $R$ be a ring. Prove that $R$ embeds in $R[X;S]$ via $a \to aX^0$ I'm not exactly sure how to approach this... I think I may need to use the fact ...
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1answer
76 views

Localization of an integer quotient is a field

Let $R:=\mathbb{Z}/24\mathbb{Z}$ be our ring, $f: \mathbb{Z}\to R$ be the canonical quotient map (i.e. $f$ sends an element to its equivalence class) and $q$ be the ideal generated by $f(3 ...
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62 views

Is quotient module finitely generated?

Suppose $R$ be any ring containing left ideal $I$. Then $I$ is submodule of $R$, so $R/I$ is R-module. My question is, is $R/I$ always a finitely generated?
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How to prove $e^{A \oplus B} = e^A \otimes e^B$ where $A$ and $B$ are matrices? (Kronecker operations)

How to prove that $e^{A \oplus B} = $$e^A \otimes e^B$? Here $A$ and $B$ are $n\times n$ and $m \times m$ matrices, $\otimes$ is the Kronecker product and $\oplus$ is the Kronecker sum: $$ A \oplus B ...
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228 views

Finite rings and subrings isomorphic to $\mathbb{Z}_n$

My book has proven this: Every ring with unity has a subring isomorphic to either $\mathbb{Z}$ or $\mathbb{Z_n}$. The $\mathbb{Z_n}$ case arises if the parent ring has characteristic $r>0$ I ...
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1answer
66 views

Simple $R$-module

Let $M$ be a simple $R$-module and $N=M\bigoplus M$. Then which one is true: 1) $N$ has a finite number of submodules. 2) $\operatorname{Hom}_R(N,N)$ is a division ring. 3) ...
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171 views

Maximal and prime ideals of $\mathbb{Z} \times \mathbb{Z}$

I have to find a maximal ideal of $\mathbb{Z} \times \mathbb{Z}$ , and a prime ideal that is NOT maximal. Or, essentially, I want $I$ such that $\mathbb{Z} \times \mathbb{Z} / I$ is a field, and I ...
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4answers
140 views

Is there a (f.g., free) module isomorphic to a quotient of itself?

My question is as in the title: is there an example of a (unital but not necessarily commutative) ring $R$ and a left $R$-module $M$ with nonzero submodule $N$, such that $M \simeq M/N$? What if $M$ ...
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2answers
67 views

Help With This Ring

Let ${\mathbb Q}[\sqrt 2, \sqrt 3]$ denote the smallest subring of ${\mathbb R}$ which contains ${\mathbb Q}, \sqrt 2$ and $\sqrt 3$. Show that this consists exactly of the real numbers of the form ...
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28 views

Notation Question regarding Ring-mod-Number and Ring-mod-Some Kernel

I'm having trouble linking the notation of something like $\mathbb{Z}/n$ and $R/H$ where $n \in \mathbb{Z}$, $R$ is a ring, and $H$ is the kernel of some homomorphism from that ring to another. In ...
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1answer
73 views

associated $\mathbb{C}[t]$- module is cyclic iff cyclic vector exists

I'm stuck on a part of a question: if $T : V \rightarrow V$ is a linear endomorphism of a $\mathbb{C}$-vector space $V$, then the associated $\mathbb{C}[t]$- module is cyclic (that is $V ...
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1answer
191 views

Height unmixed ideal and a non-zero divisor

Let $R$ be a commutative Noetherian ring with unit and $I$ an unmixed ideal of $R$. Let $x\in R$ be an $R/I$-regular element. Can we conclude that $x+I$ is an unmixed ideal? Background: A ...
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1answer
90 views

Find a basis of E as a vector space over $ \mathbb{Q} $

Find a basis for the factor ring $$\frac{\mathbb{Q}}{<16x^4-30x^3+15x^2+6>} $$ as a vector space over $\mathbb{Q} $. I honestly don't even know how to start this :( I though I would use ...