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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2answers
113 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
27 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 ...
0
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0answers
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'+...
1
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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 $\...
2
votes
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\...
1
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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 ...
1
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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\...
0
votes
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} \...
0
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0answers
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 $\mathbb{Z}[x]$ ?
3
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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?
2
votes
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
99 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 ...
2
votes
1answer
33 views

Is every element of a finite dimensional commutative non-unital $\mathbb{R}$-algebra nilpotent?

Considering a few examples of finite dimensional non-unital algebras over the reals, I tried coming up with an example of such an algebra with non-nilpotent zero divisor elements. In all the examples ...
3
votes
1answer
74 views

On graded Artinian Gorenstein algebras

Let $k$ be a field and $R$ an $\mathbb{N}$-graded $k$-algebra that is graded-commutative. Assume that $\dim_k R<\infty$ and that $R$ is Gorenstein (i.e. the injective dimension of $R$ over itself ...
0
votes
1answer
45 views

Show that $R = \mathbb{Z}[\sqrt{-17}]$ is not a Euclidean ring.

Show that $R = \mathbb{Z}[\sqrt{-17}]$ is not a Euclidean ring. To do this I tried showing that the ring is not a principal ideal domain. I wonder if this is enough and how to actually show that it is ...
0
votes
2answers
60 views

Help with understanding quotient ring structure

Let $R$ be the ring $\mathbb{Z}[x]/((x^2+x+1)(x^3+x+1))$ and $I$ be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$? I am having a hard time understanding what the ring ...
-1
votes
2answers
26 views

Definition of irreducible element of a ring

I found in my notes the following definition: Let $r\neq 0$, $r$ non-invertible. $r\in R$ is called irreducible iff $r=a\cdot b$ with $a,b\in R$ then either $a\in U(R)$ or $b\in U(R)$. Why does ...
1
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1answer
42 views

Proving $\Bbb Z/p\Bbb Z\cong R/pR$

Let $R$ be a ring of square-free order $n$. If $p \mid n$ then $\Bbb Z/p\Bbb Z\to R/pR$ is a well-defined isomorphism. I'm really unsure how to approach this problem. So we need to show that if $...
0
votes
1answer
45 views

Certain Subset of a Ring [closed]

Does there exist an infinite ring $R$ with finitely many units and an infinite $S \subseteq R$ \ $\left\{0 \right\}$ such that: There exists finite $X \subseteq S$ such that for every $y \in S$, ...
7
votes
4answers
416 views

Is my understanding of quotient rings correct?

Amidst all the rigorous constructions of quotient rings involving equivalence relations and ideals, I feel that I have finally grasped what a quotient ring is. I have applied this intuition to a few ...
0
votes
1answer
32 views

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible.

For a field $K$, show that $f(x)=x^4+x^2+1\in K[x]$ is not a unit and not irreducible. What I tried: To show that $f$ is not a unit I did the following. Suppose that $f$ is a unit, then there exists ...
2
votes
2answers
50 views

Question on splitting field and irreducible polynomials.

Let $K$ be a field, and consider a monic irreducible polynomial $f(x) \in K[x]$. Denote $d = \deg(f)$, and let $g(x) = f(x^2)$. Furthermore, let $\alpha \in \Omega^g_K$ (the splitting field of $g$ ...
0
votes
0answers
32 views

Taylor's expansion in polynomial ring

While looking proof of Hilbert's Nullstelllensatz in M. Artin's Algebra, I faced some problems. See below: The last expression for $f(x)$ is called in the book as Taylor's expansion. ...
2
votes
1answer
40 views

Books about multivariate polynomials

I'm looking for a book on multivariate polynomials, preferably a monograph (could also be a chapter inside another book). I'm interested in what can be said about roots, factoring, irreducibility, ...
3
votes
2answers
71 views

Ring of order $n$ is isomorphic to $\Bbb Z/n\Bbb Z$, with $n$ square-free

Let $R$ be a ring of order $n$ and suppose $n$ has no square in its prime decomposition. How do I see that $R$ is isomorphic to $\Bbb Z/n\Bbb Z$? I bet that the map $\Bbb Z \to R, \, 1\mapsto 1_R$ ...
1
vote
2answers
67 views

Ring with four solutions to $x^2-1=0$

I am looking for a ring $R$ in which $2$ is invertible and there are four solutions to $x^2-1=0$. $R=\Bbb Z/8\Bbb Z$ has the four solutions $1,3,5,7$ to $x^2-1=0$, but $2$ is not invertible.
6
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0answers
143 views

Can we characterize all infinite Euclidean-domains having exactly one invertible element?

$\mathbb Z_2$ and $\mathbb Z_2[x]$ are two euclidean-domains having exactly one invertible element ; my question is ; Can we characterize all euclidean domains $D$ having exactly one invertible ...
1
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1answer
30 views

unit and proper divisor of zero in a ring

unit and proper divisor of zero A unit in a ring R cannot be a proper divisor of zero. Let x ∈ R be a unit. Hence there exists a y ∈ R such that x · y = y · x = 1. Suppose x · w = z for some w ∈ R.(...
0
votes
1answer
47 views

How to show the dimension of the vector space K[X]/fK[X]?

Let K be a field and f$\neq$0 $\in$ K[X] a polynom. a) Show that the Ring K[X]/fK[X] is a K-vector space with the dimension n=deg(f) b) f is called irreducible, if for g,h \in K[X] we have f=g*h $\...
2
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1answer
28 views

Singular ideal containing a given nilpotent ideal

Let $R$ be a ring with identity, and $Z(R_R)$ be the singular ideal. Is it true that any nilpotent ideal of $R$ lies in $Z(R_R)$? It is well known that any central nilpotent element would belong to ...
1
vote
2answers
52 views

Not fully understanding polynomial quotient rings.

This is my (informal) understanding of a quotient ring. I understand that this is very flimsy, but I hope you can get the main idea. You have some ring $R$ and you want to quotient out an ideal $I$...
0
votes
1answer
89 views

If $R$ is an integral domain and $R[x]$ is an euclidean domain, then $R$ is a field [closed]

Is this obvious? I cannot see that this is true. The converse is fairly obvious though. I tried to show $(x)$ is a maximal ideal and try the quotient but failed. I will appreciate any help.
0
votes
1answer
61 views

If $A$ is an integral domain with a finite number of primes then $Q(A)=A_a$ for some $a \in A$ [closed]

If $A$ is an integral domain with a finite number of prime ideals is it possible to get the field of fractions localizing only by a set $\{a^k\}$?
3
votes
0answers
44 views

Finding units in quadratic integer rings

I want to find the units in $\mathbb{Z}[\alpha]$, where $\alpha=\frac{1+\sqrt{-11}}{2}$. One can of course use norms to find the units in quadratic integer rings of the form $\mathbb{Z}[\sqrt{D}]$ ...
0
votes
1answer
24 views

Restricting the quotient map of rings to a subring

When $q$ maps $R$ to $R/I$ and $p$ is the restriction of $q$ to a subring $A$ of $R$, why is the image of $p$ $(A+I)/I$? $q$ maps $r$ to $r+I$, so shouldn't $p$ map $a \in A$ to $a+I$, so image of $p$...
3
votes
1answer
66 views

Bijection between compact space $K$ and maximal ideals of real-valued functions on $K$

Let $K$ be a compact topological space, and denote by $R$ the ring of continuous functions $K \to \mathbb{R}$, with addition and multiplication defined pointwise. We prove that there is a bijection ...
0
votes
2answers
55 views

Cardinal of quotient rings of gaussian integers. [duplicate]

It is known that $\mathbb{Z}[i]$ is a PID and that $\mathbb{Z}[i]/(a+bi)\mathbb{Z}[i]$ is finite for all $(a,b) \in \mathbb{Z}^2\backslash \{(0,0)\}$. My question : Is there any result on the ...
1
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0answers
55 views

Converse to Chinese remainder, domain, isomorphism as rings

The converse to CRT asks: does $R/(I \cap J)\simeq R/I \times R/J$ imply $I+J=R$? For me $\simeq$ is an abstract ring isomorphism, sending $1$ to $1$, not necessarily $R$ linear, i.e., one of $R$--...
0
votes
2answers
22 views

Division ring if and only if it has no proper left ideals.

Let $R$ be an associative ring with $1$. Prove that $R$ is a division ring if and only if $R$ has no proper left ideals. Clearly, if $R$ is a division ring and $I\neq\{0\}$ is a left ideal, then ...
1
vote
1answer
47 views

Any subring of $A$ is an ideal. If $A$ is an integral domain then $A$ is commutative

Any subring of $A$ is an ideal. If $A$ is an integral domain then $A$ is commutative. Is my proof correct? So let $a$ and $b$ nonzero elements of $A$. $C(a)=\{ x\in A \mid ax=xa\}$ Is a subring ...
0
votes
3answers
66 views

Show that $R_P$ has a unique maximal ideal

Problem is: Let $R$ be a commutative ring and let $P$ be a prime ideal. (a) Prove that the set of non-units in $R_{P}$ is the ideal $P_{P}$. (b) Deduce that $R_{P}$ has a unique ...