This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

learn more… | top users | synonyms (2)

4
votes
3answers
123 views

$\mathbb{Z}[X]/(2x+4,x^2-3) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ [duplicate]

Someone has asked a question before regarding the ring $\mathbb{Z}[X]/(2x+4,x^2-3)$, but the answer wasn't quite what I was looking for. I was wondering how one would show this quotient isomorphic to ...
1
vote
2answers
107 views

Why would a field *not* be considered a discrete valuation ring?

There are two theorems in Matsumura (p. 78-9) Theorem 11.1 Let $R$ be a valuation ring. Then the following conditions are equivalent: (1) $R$ is a DVR (2) $R$ is a PID (3) $R$ is ...
1
vote
2answers
53 views

Show that the rings $R_1=F_5[x]/(x^2)$ and $R_2=F_5\times F_5$ are not isomorphic.

Show that the rings $R_1=F_5[x]/(x^2)$ and $R_2=F_5\times F_5$ are not isomorphic.($F_5$ is the field with $5$ elements.) My Work: Since $(0,1)$ does not have an inverse, $F_5\times F_5$ is not a ...
0
votes
1answer
163 views

Prove that some canonical homomorphism is injective.

Let $A \not= \{0 \}$ be a Noetherian commutative ring and let $M$ be an $A$-module. Prove that the canonical homomorphism $$M \to \bigoplus_{P \in \text{Ass}(M)} M_p$$ is injective. My question is, ...
1
vote
1answer
62 views

Representation of Algebraic Extensions by Matrices

Let $\mathbb C$ be the field of complex numbers and $\mathbb R$ the field of real numbers. It is well known that the field $\mathbb C$ can be represented as $$\mathbb ...
3
votes
0answers
85 views

Can $\operatorname{Spec}(A)$ be expressed as an inverse limit?

We know that given a ring $A$ such that $A/\mathfrak{R}$ is absolutely flat, then $\operatorname{Spec}(A)$ is Hausdorff (it's an equivalence). So $Spec(A)$ becomes a quasi-compact, Hausdorff and ...
1
vote
1answer
160 views

How to find algebraic connections between zeros of a polynomial?

Let $f(x)$ be an irreducible integer polynomial of degree $k$. Let $x_1,x_2,...,x_j$ be some zeros of $f(x)=0$ where $j<k$. How do I find identities of type $P(x_1,x_2,...,x_j) = 0$ where $P$ is ...
0
votes
1answer
54 views

Any finite integral domain is a field [duplicate]

I'm going through the Rings section of Abstract Algebra by Dummit and Foote, and I have a question about an early Corollary's proof. Proof: Let R be a finite integral domain and let a be a nonzero ...
0
votes
1answer
143 views

Zero dimensional Gorenstein ring

Let $(R,\mathfrak m)$ be a zero dimensional Gorenstein ring and $\mathfrak q$ be an $\mathfrak m$-primary ideal of $R$. Then TFAE: 1) $\mathfrak q$ is irreducible, 2) $(0:\mathfrak q)$ is principal, ...
0
votes
1answer
106 views

meadows and fields, aren't $0^{-1}=0$ can be proven simply from the axioms of fields?

Recall field axioms In this article http://www-compsci.swan.ac.uk/~csjvt/JVTPublications/RationalsAsADT.pdf Page 4, we have the SIP \begin{matrix} \left(-x\right)^{-1}=-\left(x^{-1}\right) \\ ...
1
vote
1answer
93 views

A nonprincipal ideal and a nonprime irreducible in $\mathbb{Z}[\sqrt{-17}]$

The problem asks to find a nonprincipal ideal and a nonprime irreducible in $R = \mathbb{Z}[\sqrt{-17}]$. Since $-17 \equiv 3 \pmod 4$, $R$ is the ring of integers of $\mathbb{Q}(\sqrt{-17})$. I ...
1
vote
4answers
125 views

Find a “simpler description” for $\mathbb{Z}[X]/(X-5,X^2+3)$

The problem asks for a "simpler description" of the ring $\mathbb{Z}[X]/(X-5,X^2+3)$. I could use the Chinese Remainder Theorem if $\mathbb{Z}$ were replaced by $\mathbb{Q}$, but here the ideals ...
4
votes
3answers
210 views

Can the complex numbers be realized as a quotient ring?

Can the complex numbers be realised as some $R/M$ where $R$ a ring and $M$ a maximal ideal like the integers modulo some prime? I understand that unlike the latter case, such a maximal ideal would ...
4
votes
2answers
111 views

finitely generated subgroup of $\mathbb{Q}^n$

For such a seemingly standard problem, I can't seem to find a reference for it... Prove that if $A$ is a finitely generated subgroup of $\mathbb{Q}^n$ then it has the form $\{\sum_{i=1}^k n_ia_i:n_i ...
0
votes
1answer
53 views

Tensor product of quotient and kernel

In my problem I have a PID $R$, elements $0\neq a,b\in R$ and a map $\phi_a:R\rightarrow R$ where $r\mapsto ar$. Assuming I have done all my previous calculations right I need to prove that ...
0
votes
2answers
145 views

$\mathbb{Z}\bigl[1+\sqrt{-7}\bigr]$ is not Euclidean

The similarly looking ring $\mathbb{Z}\Bigl[\dfrac{1 + \sqrt{-7}}{2}\Bigr]$ actually is Euclidean (see here). Now I want to show that $\mathbb{Z}\bigl[1 + \sqrt{-7}\bigr]$ is not Euclidean. ...
0
votes
0answers
21 views

Element in distinct prime ideal

Let $K$ be a number field and $\mathcal O$ its ring of algebraic integers. Let $\mathfrak p_1,\ldots , \mathfrak p_n$ prime ideals of $\mathcal O$ with $\mathfrak p_i\neq\mathfrak p_j$ if $i\neq j$. ...
2
votes
1answer
67 views

Determine the center of ring of differential operators with coefficients in $\mathbb{C}[z_1,z_2]$

My goal is to determine what is the center of a ring $R$ generated by differential operators $z_i \frac{\partial}{\partial z_j}$ for $i,j \in \{1,2\}$ with coefficients in polynomial ring ...
1
vote
1answer
109 views

Prove, that if the commutative ring has no zero divisors, then it is a field [duplicate]

Let $R$ be a commutative finite ring in which $ab = 0$ implies either $a = 0$ or $b = 0$ for any $a,b \in R$. Then, $R$ is a field. I do not understand how I should act. I tried different ways, but ...
1
vote
1answer
65 views

Suppose $R=F$ is a field. Prove that an $R-$module $M$ is Artinian iff it's Noetherian iff $M$ is a finite dimensional vector space over $F$.

Suppose $R=F$ is a field. Prove that an $R-$module $M$ is Artinian iff it's Noetherian iff $M$ is a finite dimensional vector space over $F$. If $M$ is a finite vector space over $F$, then neither do ...
1
vote
1answer
77 views

Endomorphisms of a ring

Let $R$ be a ring with identity and let $R^n=P⊕P'$ be a direct sum decomposition with right $R$-modules as its components. We take $e\in\operatorname{End}(R^n_R)$ as the projection of $R^n$ onto $P$, ...
1
vote
3answers
142 views

Question about localization

If $A \not= \{0 \}$ is a commutative ring and $P \subset Q$ are prime ideals of $A$ then of course $P \cap (A \setminus Q) = \varnothing$ so that $PA_Q = S^{-1}P$ is a prime ideal of $A_Q$ where $S=A ...
0
votes
1answer
47 views

An ordered group $G$ is Archimedean if and only if the following holds…

Let $G$ be an ordered group; then $G$ is Archimedean if and only if the following condition holds: $$\text{if} \space a, b \in G \space \text{with} \space a>0, \space \text{ there exists a ...
2
votes
4answers
447 views

Unit Ideal and its generators

Let $R$ be a commutative ring with unit and let $a,b\in R$ be two elements which together generates the unit ideal. Show that $a^2$ and $b^2$ also generate the unit ideal together. My Work: Unit ...
0
votes
2answers
73 views

Automorphisms in $\mathbb{R}$

Let $\phi: \mathbb{R}\rightarrow \mathbb{R}$ be an automorphism. Suppose $p=\frac{m}{n}$ is a rational number. Then is it true that $\phi(p)=\frac{\phi(m)}{\phi(n)}$? I got this problem while doing ...
1
vote
1answer
76 views

Integral closure of a PID is torsion free

Can anyone explain me why the integral closure of a PID $A$ in a separable finite extension of its fraction field is a torsion free $A$-module? I know that it is a finitely generated A-module ...
2
votes
1answer
107 views

Equivalent conditions for an ideal to be prime

Let $R$ be a commutative ring. An ideal $I$ is called prime if whenever $ab\in I$ then $a\in I$ or $b\in I$. I want to show that $I$ is prime if whenever $JK\subseteq I$, then $J\subseteq I$ or ...
5
votes
1answer
99 views

Does $(X)(Y)=(XY)$ for $X,Y\subseteq R$?

Let $R$ be a commutative ring. Denote by $X\ast Y=\{xy\mid x\in X,y\in Y\}$ the complex product of subsets. I want to show that given subsets $X,Y\subseteq R$ the following ideals are equal: ...
4
votes
1answer
126 views

Finite integral domains are commutative?

Here, integral domain is a non-zero ring $R$ (not necessarily commutative, and not necessarily contains unity), in which $ab=0$ implies $a=0$ or $b=0$. Question If $R$ is a finite integral domain, is ...
2
votes
1answer
82 views

Prove that for all $x \in R$, the ideal $xR$ is proper.

Let $R$ be a commutative ring without identity. Suppose $R$ doesn't contain a proper maximal ideal, and $R$ is not the zero ring. Prove that $\forall x \in R$, the ideal $xR$ is proper.
3
votes
1answer
211 views

Ideals and the distributive property

Does an ideal necessarily obtain the distributive property of its ring? Forgive me if the answer is obvious, I am new to ring theory. Also, any recommendation of a ring theory text would be ...
4
votes
1answer
285 views

Local Noetherian domain of dimension one with principal maximal ideal

Let $(A,\mathfrak{m})$ be a local Noetherian domain of dimension one and suppose that $\mathfrak{m}$ is principal. I wish to show that every non-zero ideal of $A$ is a power of $\mathfrak{m}$. I have ...
6
votes
1answer
150 views

Local Artinian rings with a principal maximal ideal

I would be very grateful if someone would check my proof of the following result (this is not homework). All rings are commutative and unital. Proposition: If $(A,\mathfrak{m})$ is a local Artinian ...
2
votes
1answer
55 views

Question regarding gcd in polynomial ring over a field

Let $\mathbb{F}_q$ be a finite field. We have a polynomial ring $\mathbb{F}_q[t]$ and its field of fractions, which we denote $\mathbb{K}$. Suppose I have polynomials $f_1, \ldots, f_n$ in ...
4
votes
2answers
49 views

Definition of ordered ring/field

One way to define an ordered ring is as a ring with a total order $\leq$, satisfying $x\leq y\implies x+z\leq y+z$; $0\leq x$ and $0\leq y\implies 0\leq xy$. Does it hurt to replace 1. with the ...
0
votes
1answer
29 views

Ideals of $\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$

I need to describe all the ideals of $\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$ I suppose that trivials, and $(0,..,1_{i},..,0)\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$ for any $i$ and nilradicals ...
0
votes
2answers
43 views

How can $\Bbb{Z}/10$ be viewed as a $\Bbb{Z}$-module?

How can $\Bbb{Z}/10$ be viewed as a $\Bbb{Z}$-module? For example, when I compute $5.\overline{5}$, where $5\in\Bbb{Z}$ and $\overline{5}\in\Bbb{Z}/10$, is this equal to $5$?
1
vote
1answer
47 views

Is the number of prime ideals in a ring of algebraic integers countable?

My question is: If $K$ is a number field, then $Spec(\mathcal O_K)$ is countable or non-countable.
2
votes
1answer
98 views

Integral domain without unity has prime characteristic?

By an integral domain, we mean here, a ring (not necessarily with unity) in which $ab=0$ implies $a=0$ or $b=0$. Question: If an integral domain without unity has positive characteristic, is it ...
0
votes
1answer
100 views

Height and coheight of an ideal

Given an ideal $\mathfrak{a}$, Matsumura defined the height of $\mathfrak{a}$ as: $$\text{ht}(\mathfrak{a})=\inf_{\mathfrak{p}\in V(\mathfrak{a})}\text{ht}(\mathfrak{p})$$ He states that: ...
4
votes
2answers
142 views

About rings with a right multiplicative identity $1$ such that every element in $R\setminus\{0\}$ has a left-inverse

$R$ be a ring with a right multiplicative identity $1$ such that for every $a \in R \setminus \{0\}$, $\exists x \in R$ such that $xa=1$ i.e. every element in $R \setminus \{0\}$ has a ...
3
votes
1answer
138 views

Irreducible and prime elements

In my commutative algebra lecture notes it says: A non-zero element $p$ of a ring $R$ which is not a unit of $R$ is called a prime element if $p=ab$ implies $a$ is a unit or $b$ is a unit. Is this ...
4
votes
3answers
112 views

Does there exist a unital ring whose underlying abelian group is $\mathbb{Q}^*$?

Let $\mathbb{Q}^*$ be the group of units of the rational numbers. Does there exist a unital ring whose underlying additive group is $\mathbb{Q}^*$? I don't really have a gut feeling yea or nea. ...
1
vote
1answer
129 views

judge if nilradical equals jacobson radical

judge if nilradical equals jacobson radical 1)a noetherian ring that is not a artin ring. 2)a local integral domain that is not a field. 3)a integral domain with only finite number of ...
1
vote
1answer
101 views

Example of non noetherian ring and noetherian $\Bbb Z$-module

a non Noetherian ring that is a Noetherian $\Bbb Z$-module a Noetherian ring that is a non Noetherian $\Bbb Z$-module I have no idea in 1, and I'm not sure if $\mathbf{Q}$ is right for 2? ...
0
votes
1answer
23 views

$R$ is the ring $\mathbb{Z}[\sqrt{-k}]$. In $R$, if $3\mid (a+b\sqrt{-k})$, then $3\mid a$ and $3\mid b$ in $\mathbb{Z}$

$R$ is the ring $\mathbb{Z}[\sqrt{-k}]$ where $k\ge 5$ and $k\equiv2 \pmod{3}$. I would like to prove that in $R$, if $3\mid(a+b\sqrt{-k})$, then $3\mid a$ and $3\mid b$ in $\mathbb{Z}$. I have ...
0
votes
2answers
206 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every proper ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is ...
0
votes
2answers
47 views

Example of ideals such that $I^n=0$ but $I^{n-1}\not= 0$

Let $R$ be a ring. For each $n>0$ I want to find an ideal $I$ of $R$ such that $I^n=0$ but $I^{n-1}\not= 0$. Clearly this won't work for $R=\Bbb{Z}$ or $\Bbb{Z}/n\Bbb{Z}$. And I ran out of ...
1
vote
1answer
59 views

Are the two ways of creating an $S^{-1}A$ algebra equivalent?

Let $f:A\to B$ be a ring homomorphism and $S$ be a multiplicative set, define $S^{-1}B$ to be $B\times S$ with equivalence relation $(b,s)\sim(b',s')$ iff $\exists t\in S$ such that $t(sb'-s'b) = 0$. ...
2
votes
2answers
62 views

MCS meet all prime ideals

let A be a commutative ring, is there any multiplicatively closed subset S (not containning 0), s.t. every prime ideal in A intercept S is not empty? My thinking is that there is 1-1 ...