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

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1answer
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An induction proof on a version of “prime avoidance” from Atiyah-McDonald.

The proposition is 1.11 from the commutative algebra book. Let $\mathfrak{p}_1, \dots, \mathfrak{p}_n$ be prime ideals and let $\mathfrak{a}$ be an ideal contained in the union of those prime ideals. ...
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34 views

Construction of the sets.

At this question, how to construct $G_1$, $G_2$, $*G$. When I am constructing these sets, I make a mistake. Please show these sets explicitly. Thanks.
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31 views

Online Finite Field Calculator

I need to find an online Finite Field calculator with the following operations: Inverse SqrRoot Mult Div I have found one a couple of days ago but lost the url, and cannot find it now. Any ...
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0answers
15 views

$V(\mathfrak{a} \cap \mathfrak{b}) = V(\mathfrak{a}) \cup V(\mathfrak{b})$ (Spectrum of a commutative ring)

Let $V(\mathfrak{a})$ be all ideals in ${\rm Spec}(A)$ that contain ideal $\mathfrak{a}$. Then $V(\mathfrak{a} \cap \mathfrak{b}) = V(\mathfrak{a}) \cup V(\mathfrak{b})$. $\mathfrak{p} \in$ RHS ...
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1answer
35 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
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0answers
43 views

Global dimension of $\mathbb Q [x]$

I'm trying to show that the global dimension of $\mathbb Q [x]$ is 1. I have shown that $D(\mathbb Q [x]) \leq 1$ as follows. One can reduce to the case of showing that $$sup_{\{B\}}\; \text{pd}\; ...
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1answer
36 views

Questions on a proof of “All prime ideals of a Dedekind domain are invertible”

I tried to prove this theorem : All prime ideals of a Dedekind domain is invertible. i.e, For every prime ideal $\mathfrak{p}$ of Dedekind domain $R$, there exists $\mathfrak{p}^{-1} \subseteq ...
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1answer
34 views

$\Bbb Z[i\alpha]$ UFD's

I know that $\Bbb Z[i]$ and $\Bbb Z[\sqrt{-2}]$ are Unique Factorization Domains, and that $\Bbb Z[\sqrt{-6}]$ is not. I have two questions. I know that they may be difficult questions, so I only ask ...
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20 views

Proving that $V(R^*)=V(R)-1$

Let $R$ be a noetherian local ring with Jacobson radical $J$. Define $V(R)=dim J/J^2$ where $J/J^2$ is considered as a vector space over $R/J$. Now fix $x\in J/J^2$. If we then let $R^*=R/xR$ then I ...
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3answers
45 views

Let $R$ be a PID and $I$ is a non zero proper ideal of $R$. show that if $R/I$ has no nonzero zerodivisor, then it is a field. [on hold]

Let $R$ be a PID and let $I$ be a non-zero proper ideal of $R$. Show that if $R/I$ has no non-zero zerodivisor, then it is a field.
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2answers
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Sub-modules of free modules

I'm going back through basic module theory notes, and I've come across a paragraph explaining that a sub-module of a finitely generated free module may not itself be free. In my course a free module ...
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1answer
37 views

Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
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1answer
55 views

Ring of fractions $S^{-1}A$ and localisation

I'd really appreciate if somebody could help me with the problem 6.4 Reid (Undergraduate commutative algebra), because I've been trying to get the solutions for days and I don't see it. (a) Give an ...
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1answer
22 views

Notation Question about Rings

If $S = \langle2\rangle$ is the ideal generated by $2$ in $\mathbb{Z}$, what does $S[x]$ represent?
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2answers
48 views

What's the point of defining left ideals?

I admit, I haven't gotten really far in studying abstract algebra, but I was always curious (ever since I saw a definition of an ideal) why is the notion of left-sided ideal introduced when we ...
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0answers
30 views

Show by explicit calculation that $\varphi\colon\mathbb{Z}\to\mathbb{Z}_n, m\mapsto m\% n$ is a surjective ringhomomophism

Consider $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Show by explicit calculation ...
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1answer
71 views

Are $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ isomorphic?

I saw somewhere that $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ considered the same. Is it true? Why? I'm a beginner so please answer in details
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16 views

Show that the solutions lifted by Hensel's lemma, are the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$

Suppose that $f(x) \in \mathbb{Z}[X]$ is a polynomial to which we can apply Hensel's Lemma. Suppose the set $A$ is the set of solutions of $f(x)$ in $\mathbb{Z} / p \mathbb{Z}$. I'd like to show that ...
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4answers
33 views

Subrings Between Integers and Rationals

I'm trying to come up with an example of a ring that is bound strictly between the integers and the rational numbers, but I'm finding this construction very difficult. If anyone has any advice on how ...
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1answer
18 views

If $l(a, b, c) = l(a', b', c')$, then $(a, b, c) = (a ', b', c')k$ for some $k \in F$?

Let $F$ be a division ring. Define $l(a, b, c) = \{(x, y, z) \in F^3 : xa + yb + cz = 0\}$. Question: If $l(a, b, c) = l(a', b', c')$ is it true that $(a, b, c) = (a', b', c')k$ for some $ k \in F$? ...
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67 views

A book suggestion - Algebraic geometry. (Arf rings and Hilbert functions)

I am studying algebraic geometry and I need to learn Arf rings and Hilbert functions. Please suggest me books / lecture notes... etc that explains this topic in detail. Thank you.
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1answer
47 views

ring and module problem

Let $$F=\mathbb{R}$$ $$V=\mathbb{R}^{4}$$ consider two matrices $$S1=\begin{vmatrix} 0&-1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& -1\\ 0&0 & 1 & 0 ...
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1answer
33 views

Isomorphism between rings

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I = \left\lbrace f \in \mathbb{R} : f^2(0) + f^2(1) = 0 \right\rbrace$. 1) Prove that $I$ is an ideal. ...
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3answers
28 views

Show that if $m\in M_n$ and $k \in \Bbb Z_n$ then $mk\in M_n$.

Let $M_n = \{a\in \Bbb Z_n\mid \text{ there exists a non-zero integer $k$ with the property that }a^k\equiv 0 \pmod n \}.$ Show that if $m \in M_n$ and $k \in \Bbb Z_n$, then $mk\in M_n$. For which ...
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0answers
49 views

$p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$

I need to show that $p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$ Here's what I've done: Please tell me if it's correct Over $\mathbb C,$ $x^4-2x^2-4\\=(x^2-1)^2-5\\=(x^2-1+\sqrt ...
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1answer
33 views

R be the ring of real valued continuous functions on closed interval [0, 1] and I an ideal of R.

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I$ be an ideal of $R$. Assume $f(x) = \cos{(2πx)}$ and $g(x) = 2x$ both belong to $I$. Does $h(x) = ...
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11 views

Are these computational models equivalent?

Let $f : X \to Y$ be a problem that you want to compute. Say we have an $O(1)$-computable maps, $\phi, \psi$, such that $X \xrightarrow{\phi} (\Bbb{Z}_n)^k \xrightarrow{\psi} Y$. After all, ...
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2answers
25 views

Self inverting Rings

Would it be possible for a ring to have elements that are their own additive inverses? What I mean is, would it be possible to have a ring $K$ of mathematical objects $A$ such that: $$A+A=i,\;\forall ...
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1answer
28 views

Definition of Direct Sum of Ideals

I've been searching the internet, and I can't find a definition for the direct sum of ideals. In a previous question I posted, the author writes $M_n(D) =\oplus I_R$, where the $I_R$ are subrings and ...
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2answers
36 views

Epimorphism affect on Ideals

Let $f:R\rightarrow S$ be an epimorphism of commutative rings with unit $1$ and let $I\unlhd R$ be a maximal ideal that does not contain $ker(f)$. I'm trying to show if it's true or not that $f(I)$ is ...
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49 views

Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
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46 views

Characteristic of Integral-domain where $15a=0$ but $3b\neq 0$.

Let $R$ be an integral domain. Let $a,b \in R$. Assume that a and b both not zeros, $15a = 0$ and $3b \neq 0$ group. What can you say about the characteristic of $R$?
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27 views

On an invertible element for equivariant K-theory

Fix a positive integer $m$. Let $G = \lbrace h\in\mathbb C | h^m = 1\rbrace$ and $(X,\pi)$ the standard representation of $G$. Namely $X = \mathbb C$ and $\pi:G \to GL(X)$ is defined by $\pi(h)v=h v$ ...
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1answer
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+100

Equivalence Of Definitions Of Prime Ideal In Ring Without $1$

Let $R$ be a rng, so that $1\not\in R$. I am trying to show that following are equivalence of definition of prime ideal $P$; i) $AB\subseteq P$ with $A,B\subseteq R$ implies $A\subseteq P$ or ...
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3answers
37 views

General question about quotient rings

I hope to better understand the notion of a quotient ring through this example: I am given $R=\mathbb{Z}[i]=\{a+bi:a,b\in \mathbb{Z}\}$ and $M=\{a+bi: 3|a,3|b\}$. I have already shown that $M$ is a ...
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2answers
77 views

Proof of Wedderburn's Theorem

I've been going through a proof of Wedderburn's theorem: and I'm stuck on the very last part, where the author refers to example 2.1.4. (linked below). I don't understand what $D^n$ means, or why it ...
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24 views

What is the definition of a norm in the context of rings?

On several places in ring-theory I encountered so-called 'norms'. For instance on integral domain $\mathbb{Z}\left[i\right]$ with prescription $a+bi\mapsto a^{2}+b^{2}$ where it also serves as a ...
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2answers
51 views

Show module is Noetherian

Let $R$ be a commutative ring and let $0 \to L \to M \to N \to0$ be an exact sequence of $R$-modules. Prove that if $L$ and $N$ are noetherian, then $M$ is noetherian. I tried considering the pre ...
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1answer
30 views

What exactly are the elements of $\mathbb{Z}_p[x]/\langle p(x) \rangle$?

It is wellknown that for a polynomial ring $\mathbb{Z}_p[x]$, $\mathbb{Z}_p[x]/\langle p(x) \rangle$ for prime $p$ is a field if and only if $p(x)$ is irreducible over the given polynomial ring, in ...
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2answers
49 views

Finitely many prime ideals lying over $\mathfrak{p}$

Let $A$ be a commutative ring with identity and $B$ a finitely generated $A$-algebra that is integral over $A$. If $\mathfrak{p}$ is a prime ideal of $A$, there are finitely many prime ideals $P$ ...
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1answer
25 views

Integral multiplicative system over a domain

Suppose $A$ is a domain and $S\subseteq A$ is a multiplicative system. Show that $S\subseteq A^\times$ if and only if $S^{-1}A$ is integral over $A$. I've started $\Leftarrow$ below... Suppose ...
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1answer
49 views

Necessary and sufficient condition for $r(\mathfrak a)$ to be prime

As we know, $$\mathfrak a~\text{is a primary ideal}\Rightarrow r(\mathfrak a)~\text{is a prime ideal}. $$ But $r(\mathfrak a)$ may not be a prime ideal if $\mathfrak a$ isn't a primary ideal. ...
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isomorphic quotient rings?

I have trouble in determining, whether two rings are isomorphic: Let's have $R = GF(3)$ and rings $R[x]/(x^2+x+2)$ and $R[x]/(x^2+2x+2)$. How can one determine whether these two rings are ...
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1answer
21 views

Find all elements of quotient ring

I am studying the definitions of rings, ideals, and quotient ring, but I have a bit problem to apply the theory into the practice. I would like to find all elements of quotient ring $\mathbb{Z}[i]/I ...
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2answers
63 views

Examples of Cohen-Macaulay rings.

I've just started to learn about Cohen-Macaulay rings. I want to show that the following rings are Cohen-Macaulay: $k[X,Y,Z]/(XY-Z)$ and $k[X,Y,Z,W]/(XY-ZW)$. Also I am looking for a ring which is ...
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Is there a characterization of integral domains in terms of the homomorphisms out of them?

In the $\mathbf{Set}$-concrete category of commutative rings, we can define that an object $A$ is a field iff for every homomorphism $f : A \rightarrow B$, precisely one of the following holds. $f$ ...
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1answer
69 views

Every radical is prime?

$a$ is an ideal of $A$. $$f:A\to A/a,\ \ x∈r(a)$$ r(a) is a prime ideal? proof 1: $x^n\in a$ for some $n \Rightarrow (x+a)^n\in a$ for some $n \Rightarrow f(r(a))=\text{nil-radical}$ in $f(a) ...
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39 views

Question about Principal Ideals

I'm just learning basic ring theory and had a question about the definition of a principal ideal. For a commutative ring $R$ with unity, Fraleigh defines the principal ideal generated by $a\in R$ as ...
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3answers
44 views

Maximal Ideal of $\mathbb{Z}[i]$

I'm trying to show that $<1-i>$ is a maximal ideal of $\mathbb{Z}[i]$. I started by assuming there is some ideal $A$ that properly contains $<1-i>$, and then I want to show that $1 \in ...
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2answers
38 views

Localization of a Dedekind domain.

I have a question on localizations of Dedekind rings which I am learning about in an undergraduate class. Let $R$ be a Dedekind ring with quotient field $K$, $\mathfrak p$ a nonzero prime ideal in ...