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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Exercise on the ring $\mathbb Z \times \mathbb Z$ and its quotient with an ideal

Let $A = \mathbb Z \times \mathbb Z$ a ring, where operations are defined elementwise. a) Prove that the ideal $I$ generated by $x = (4,6)$ is not maximal. b) Find in $A$ (if it exists) an ...
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46 views

Maximal ideals in the ring of measurable functions

The $R$ ring of continuous functions from $[0,1]$ to $\mathbb{R}$ has a property that its maximal look like a subset of $R$ consisting of those functions which vanish at a common single point in $[0,...
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2answers
44 views

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$, does that mean that $s\not\mid r$ in $R$?

Given a ring $R$ and a ring extension $R'$, if $r=r's$ where $r'\in R'\setminus R$ and $r,s\in R\setminus\{0\}$, does that mean that $s\not\mid r$ in $R$? I was thinking for example in $\Bbb{Z}$, ...
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2answers
89 views

Is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$?

I came over a question in ring theory which I am not being able to proceed upon: When is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$, where $m, n \in \mathbb{N}$? I know that to ...
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1answer
104 views

Prove that $\Bbb{R}[\cos(\theta),\sin(\theta)]\cong\Bbb{R}[x,y]/(1-x^2-y^2)$ [duplicate]

More precisely, given the ring homomorphism $\phi:\Bbb{R}[x,y]\to\Bbb{R}^\Bbb{R}$, with $\phi(f(x,y)):\Bbb{R}\to\Bbb{R},\,\,\phi(f(x,y))(\theta)=f(\cos(\theta),\sin(\theta))$, where $\Bbb{R}[x,y]$ is ...
0
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1answer
23 views

Is it true that the only regular elements in $Z_m$ are invertible ones?

I have this doubt. In a unitary and commutative ring $$Z_m = \{[0]_m, [1]_m,\ ...\ ,\ [m - 1]_m\}$$ There are only two "kind" of elements: invertible and zero divisors. Is it true to say that the ...
0
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1answer
22 views

the field $Fr (A [X])$ and $ Fr (A) (X)$ are the same

Let $A$ be an unitary integral domain , and let $Fr (A)$ its fractionary field; this field is determined (to isomorphism field) by the following universal property: a) the ring $A$ is injected by a ...
1
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1answer
32 views

A question on part of the proof of the theorem that if $R$ is a UFD then $R[x]$ is a UFD as well.

I have a question regarding a proof in Peter Falb's Methods of Algebraic Geometry in Control Theory, volume I, for the claim in the title. On pages 16-17 he proves the property (ii) of UFDs that the ...
1
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1answer
22 views

Name for submodule killed by a right ideal

Let $\mathfrak a$ be a right ideal in a ring $R$. The set $N=\{m\in M: \mathfrak am=\mathfrak 0\}$ is a submodule of the left $R-$module $M$: If $m,n\in N$, $a\in \mathfrak a$, then $a(m-n) = am-an =...
2
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2answers
274 views

Algebraic structure on any infinite set

Given any algebraic object $X$, say group, ring, integral domain, etc., and a special subset $I$ of $X$ namely normal subgroup, ideal etc., it is always possible to put a structure on $X/I$ induced ...
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1answer
42 views

Quotient ring with reducible polynomial

Let $S = \mathbb{R}[x]/(x^2+1)^2).$ The first goal is to show that there exist exactly two homomorphisms $ \pi\colon S\to \mathbb{C} $ such that $\pi|_{\mathbb{R}} = \text{id}_{\mathbb{R}}.$ I know ...
0
votes
0answers
32 views

Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
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2answers
121 views

Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
1
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1answer
33 views

$\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors

I want to show that $\mathbb{Z}[\sqrt{-5}]$ satisfies the descending chain condition of divisors: given a chain $a_1,a_2,\dots,a_n,\dots$ and $a_{n+1}\mid a_n$ for any $n\in \mathbb{N}$, then there is ...
0
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1answer
48 views

$I$ is the maximal left ideal

Let $R$ be a ring and $I\subseteq R$ the unique maximal right ideal of $R$. I have shown that $I$ is an ideal and that each element $a\in R-I$ is invertible. I want to show that $I$ is the unique ...
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2answers
72 views

Is $a$ invertible? [closed]

We have that $R$ is a ring. Suppose that $Ra=R$ and $bR=R$, for $a,b\in R$. Then we have that there is $x\in R$ such that $ab=1$ and $bx=1$. Does it follow that $a$ is invertible?
3
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0answers
80 views

Simple examples of rings from topology

The ring $C([0,1],\mathbb{R})$ of continuous functions from $[0,1]$ to $\mathbb{R}$ is an interesting example of ring due to its some interesting property (namely, structure of maximal ideals). The ...
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0answers
60 views

Which functions $\mathbb{Z} \rightarrow \mathbb{Z}$ are 'totally compatible'?

Definition 0. For each integer $k$ and each function $f : \mathbb{Z} \rightarrow \mathbb{Z}$, lets define that $f$ is $k$-compatible iff there exists a function $g : \mathbb{Z}/k\mathbb{Z} \rightarrow ...
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0answers
75 views

When does $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges?

We know $\sum_{p\in\mathbb{P}} \frac{1}{|p|^2}$ diverges where $\mathbb{P}$ denotes set of all primes in $\mathbb{Z}[i]$ (because that sum is greater that $\sum_{p \equiv 3 \mod 4} \frac{1}{p}$, which ...
0
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1answer
30 views

If $a$ is algebraic and $f\colon\mathbb{Q}[x]\to\mathbb{C}$ where $f(g(x))=g(a)$, prove that $\ker(f)$ is a maximal ideal of $\mathbb{Q}[x]$

If $a$ is algebraic, then a polynomial $p(x)$ in $\ker(f)$ is irreducible iff it generates $ker(f)$. For an ideal $I$ in $Q[x]$ containing $\ker(f)$, let $p(x)=\ker(f)$ and $q(x)=I$. Then $p$ is ...
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1answer
30 views

Number of distinct equivalence classes of $\mathbb Z_n$ of the “ associate ” equivalence relation

Define an equivalence relation on $\mathbb Z_n$ as : For $a,b \in \mathbb Z_n $ , $a\sim b$ iff $\exists k \in U_n=\mathbb Z_n^{\times}$ such that $a=kb$ (i.e. $a,b$ are related if they are "...
0
votes
3answers
39 views

If $a$ is algebraic, prove that there is a minimal polynomial $p(x)$ in $Q[x]$ such $p(a)$ = $0$.

If $f_a$: $Q[x]$ -> $C$ is the evaluation at $a$ map, then a polynomial $q(x)$ in $ker(f_a)$ is irreducible iff it generates $ker(f_a)$. Let $ker(f_a)$ = $h(x)$ so that $h(x)$ is irreducible and $f_a(...
0
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1answer
17 views

Systems with Principle Roots of Unity

Over the complexes, it's possible to have a principle root of unity - in other words, a value $\omega$ with $\omega^n = 1$, and satisfying: $$\sum_{i=0}^{n-1}{ \omega^{ij} } = 0, j \in \{1, 2, \dots, ...
0
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1answer
38 views

Showing an isomorphism of rings

Consider the ideal $I=(1+2x)\cdot \Bbb Z[x]$ in the polynomial ring $\Bbb Z[x]$. I am trying to show that $\Bbb Z[x]/I$ is isomorphic to $R=\{\frac{a}{2^r}:a\in \Bbb Z, r\in \Bbb N_0\}$. My approach:...
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2answers
48 views

Why $\langle x^2+1\rangle$ is not prime in $\mathbb{Z}_2[x]$?

I am reading ring theory (a beginner) and I stumbled upon a problem which I can't understand The ideal $\langle x^2+1\rangle$ is not prime in $\mathbb{Z}_2[x]$, since it contains $(x+1)^2=x^2+2x+...
3
votes
2answers
89 views

Factor ring and prime elements

My task is to find prime elements of the ring $\mathbb Z[\sqrt{-21}]$ and describe the factor ring $\mathbb Z[\sqrt{-21}]/(2+\sqrt{-21})$. I think that to describe factor-ring i need to find the ...
1
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1answer
69 views

Subfields of $\mathbb{C}$ which are connected with induced topology

The ring of continuous functions on $[0,1]$ to $\mathbb{R}$ has an interesting property: every maximal ideal of this ring is the subset of all functions vanishing at a common point. If we ...
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0answers
67 views

Nonprincipal lattice

Let $\Lambda$ be a $J$-module generated by the elements $v_1, \ldots, v_n$ which are linearly independent over $\mathbb{C}$, $v_i \in \mathbb{C}^n$. It is said that in case $J$ is not a PID (for ...
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1answer
83 views
1
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2answers
112 views

Describe the structure of factor ring $\mathbb{Z}[\frac{1+\sqrt{5}}{2}]/(2+\sqrt{5})$. [closed]

I'm really confused with this question... I know, that $ \mathbb{Z}[\frac{1+\sqrt{5}}{2}]=\left \{ \frac{a + b\sqrt{5}}{2} \enspace | \enspace a,b \in \mathbb{Z}, \enspace a\equiv b\pmod 2 \...
1
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1answer
26 views

$R/\langle p^k\rangle$ is an associator (i.e. if $\langle a\rangle = \langle b\rangle,$ then $a$ and $b$ are associates) when $R$ is a PID.

As the title says, I want to show that when two principal ideals are equal in $R/\langle p^k\rangle,$ where $R$ is a principal ideal domain and $p\in R$ is a prime element, then their generators are ...
0
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1answer
41 views

Interpretation of certain things in $Z_7$

$Z_7$ is the ring of integers modulo $7$. I am beginner in ring theory and a question which says Find a reasonable interpretation for the expessions $1/2\ ,\ -2/3\ ,\ \sqrt{-3}\ \&\ -1/6$ in $...
2
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0answers
60 views

Prime elements of ring $\mathbb{Z}[\sqrt{-21}]$ [closed]

Find prime elements of the ring $\mathbb{Z}[\sqrt {-21}]$. Please help with some ideas.
3
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1answer
43 views

About definition of UFD

On Wikipedia, UFD is defined as an integral domain in which every element can be uniquely factored as product of primes (irreducibles), up to multiplication by units and arrangement. My question is ...
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28 views

Singular ideal of an idealization

Let $S$ be a commutative ring, and let $A$ be a faithful $S$-module. Through idealization, we can make the abelian group $R=S⊕A$ into a commutative ring using the multiplication $(s,a)(s',a')=(ss',sa'+...
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1answer
43 views

Meaning of $Z\oplus Z$

I am a beginner in Ring Theory and just started Integral Domains. In my textbook, the following was stated : $Z\oplus Z$ is not an integral domain. I can't understand this. I know $\oplus$ ...
3
votes
1answer
68 views

Relatively prime elements in $\mathbb{Z}[i]$

I was solving the following problem: given $a+bi\in\mathbb{Z}[i]$, how many elements are there in $\mathbb{Z}[i]/(a+bi)$. I was trying to solve it in the following way: Assume $gcd(a,b)=1$ in $\...
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1answer
49 views

$k\left[x,y\right]$ is not integral over the $k\left[xy,y\right]$

I want to prove that the polynomial ring $k\left[x,y\right]$ is not integral over the subring $k\left[xy,y\right]$ , where $k$ is a field. My claim is that $x$ is not integral over $k\left[xy,y\...
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1answer
48 views

Can I deduce $f(\textrm{Ker}(g))=g(\textrm{Ker}(f))=0$ from this data?

Let $R$ be a commutative ring with identity and $A, B$ two $R$-algebras. Consider $f, g: A\longrightarrow B$, $h:B\longrightarrow A$ and $\imath:A\longrightarrow A$ morphisms of $R$-algebras ...
0
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0answers
29 views

Proving that there exists $a\in R$ such that $a \equiv a_k \pmod{I_k}$ [duplicate]

Let $I_1,...,I_m$ be ideals of a ring $R$ such that $I_j+\cap_{k\neq j}I_k=R$ for every $j\in\{1,...,m\}$. Then if $a_1,...,a_m\in R$ there exists $a\in R$ such that $a \equiv a_k \pmod{I_k}$ for ...
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1answer
63 views

Rings of Krull dimension one

I have to write a monograph about commutative rings with Krull dimension $1$, but I can't find results, so I am looking foward for some references, and some results to search. Also, I would appreciate ...
2
votes
2answers
63 views

Proving/Disproving $M$ has the structure of an $R$-module

Given an abelian group $M$ and a ring $R$, how can one prove or disprove that $M$ has the structure of an $R$-module? When proving $M$ is an $R$-module, if it is not obvious how to define an action $R\...
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1answer
34 views

Induced homomorphism on Spectra of rings

In Matsumura textbook, there is this following statement. A ring homomorphism $f:A \to B$, induces a map $f': \operatorname{Spec}B \to\operatorname{Spec}A$ under which an element $\mathfrak{p} \...
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22 views

Ring of smooth functions on a manifold and localization with respect to a multiplicative system

Take $X$ a smooth manifold and $x\in X$. It can be shown that the germ of smooth functions around $x$, $C^\infty(X)_x $ is equal to the algebraic $S^{-1}C^\infty (X)$ where $S$ is the set of smooth ...
0
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2answers
41 views

Why is $I[x]$ not maximal $\mathbb{Z}[x]$? [duplicate]

We have that $I=(2)$ is maximal in $\mathbb{Z}$ because $(2)\subseteq (4)\subseteq \dots \subseteq (2^k)$, right? Why is $I[x]$ not maximal $\mathbb{Z}[x]$ ?
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0answers
44 views

$\mathbb{Z}[\sqrt{10}]$ is noetherian

How can we prove that $\mathbb{Z}[\sqrt{10}]$ is noetherian except by using Hilbert basis theorem? How can we find a sequence of ideals that satisfy the ACC?
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1answer
19 views

Ideal generated by two irreducible polynomials is the field itself

The question is: Let $F$ be a field and $f(x),g(x) \in F[x]$. Verify that $$N=\{r(x)\ f(x)+s(x)\ g(x):r(x),s(x)\in F[x]\}$$ is an ideal of $F[x]$. Then show that if $f(x)$ and $g(x)$ have different ...
0
votes
0answers
27 views

Prove that the radical of an ideal is an ideal

Let $R$ be a commutative ring with unity. For an ideal $I$ of $R$, I am attempting to prove $\sqrt{I}=\{x\,|\,x^n\in I\}$ is an ideal. Closure under multiplication with $R$ seems straight forward: ...
5
votes
3answers
98 views

Let $R$ be a commutative ring, $\phi :R\to S^{-1}R, \phi(r)=\frac{r}{1}$ then $\phi(r)$ is invertible iff $r\in S$

$R$ is an arbitrary commutative ring with identity, and $S\subset R$ is multiplicative. I read that the map $\phi :R\to S^{-1}R, \phi(r)=\frac{r}{1}$ is characterized by the set $S'=\{s:\phi(s)\text{ ...
5
votes
1answer
42 views

When is the norm of a number even?

In the ring $$\textbf{Z}[i],$$ if the norm of an element is divisible by $2$, then the element must be divisible by $$1 + i,$$ and vice versa. A similar result holds for $$\textbf{Z}[\sqrt 3]$$ and ...