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

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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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2answers
50 views

co-idempotents: algebraic dual of an idempotent element?

So many times you can write out the axioms for an algebraic structure (say an algebra over a ring) as commutative diagrams and then reverse all the arrows and get a new structure: say a coalgebra. ...
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1answer
15 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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2answers
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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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38 views

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

I am studying algebraic geoemtry. And I need to learn Arf Ring & Hilbert funciton. Please suggest me books / lecture notes...etc. that explains this topic in detail. Thank you.
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18 views

prove that commuting ring is a subring of E(M) .

$E(M)$: the collection of all endorrmorphisms of the additive group of $M$ since $E(M)$ is redered a ring by deferring the sum and the multiplication or product, I thinked easy prove. but I don't ...
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0answers
25 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
32 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
26 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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2answers
71 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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3answers
27 views

If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.

I'm struggling with this. I know it is going to use Lagrange's Theorem so this is what I have so far: Suppose $|U_n| = k$ This implies $a^k = 1$ for all a in $U_n$ and $|a|$ divides $k$. Now, what ...
3
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0answers
46 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
31 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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0answers
9 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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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
24 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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2answers
28 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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2answers
44 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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0answers
40 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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0answers
23 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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0answers
67 views

What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $ A $ by generators and relations helps in the study of structure of the algebra ...
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0answers
14 views

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
32 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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0answers
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
47 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
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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0answers
48 views

About a class of commutative rings that they have maximal ideals for any element non-inversible in $ZF\neg AC $

Let $\mathcal{N}{oetherian}\mathcal{C}\mathcal{R}{ng} \overset{def}{=} {\left\lbrace{ R \in \mathcal{C}\mathcal{R}{ng} \wedge R \,\text{is}\, \mathcal{N}{oetherian} }\right\rbrace}$. I define the ...
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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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6answers
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Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong ...
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4answers
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Consider the ideal $\langle 1+i \rangle$ in $\mathbb{Z}[i]$.

Consider the ideal $\langle 1+i \rangle$ in $\mathbb{Z}[i]$. $(a)$ Make use of the given description of this ideal, $\hspace{75pt}$ $\langle 1+i \rangle = \{a+bi:a+b \text{ is even}\}=\{\alpha\in ...
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2answers
89 views

Prove or disprove that the ideal $(2+4\mathbb{Z},x)$ is a principal ideal in $(\mathbb{Z}/4\mathbb{Z})[x]$

Prove or disprove that the ideal $(2+4\mathbb{Z},x)$ is a principal ideal in $(\mathbb{Z}/4\mathbb{Z})[x]$. I know $\mathbb{Z}/4\mathbb{Z}$ is not a field. have something help?
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3answers
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Let $R = \mathbb Z[i]$. Show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$.

Let $R = \mathbb Z[i]$, $z = 3+i$ and $I = \langle z \rangle$. I need to show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$ and $10 \mid a$, ...
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1answer
34 views

Determining whether $2x ^ 3 - x^2 + 1$ irreducible in $\mathbb{Q}[x]$?

I'm checking if $f(x) = 2x ^ 3 - x^2 + 1$ is irreducible in $\mathbb{Q}[x]$. We have a polynomial of degree three so if it has no roots it is irreducible. But as we are dealing with $\mathbb{Q}[x]$ ...
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1answer
24 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
83 views

Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$?

Is $\mathbb{Z}[x]/(x^2+1)$ isomorphic to $\mathbb{Z}[i]$? My attempt is that try to define a mapping $g$ from $\mathbb{Z}[x]$ to $\mathbb{Z}[i]$ by $g(f(x))= f(i)$, for $f(x)\in\mathbb{Z}[x]$. ...
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3answers
41 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
60 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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2answers
54 views

“Primeness” of C[x] in B[x], where A is a subring of B and C is the integral closure of A in B.

Let A be a subring of B, and C the integral closure of A in B. If f, g are monic polynomials in B[x] such that fg is in C[x], then f, g are in C[x]. The first part of the problem allowed the ...
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1answer
68 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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2answers
33 views

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
20 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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0answers
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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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0answers
38 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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2answers
37 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 ...
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0answers
49 views

Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
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1answer
95 views

$k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$

This is part of an exercise from Eisenbud: $k$ is a field, describe as explicitly as possible a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$ b) $k[x] \otimes_{k} k[y]$ Any hint ?
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0answers
46 views

Can someone explain to me this answer about subrings?

so I know how to prove that $\mathbb{Z}\left[\sqrt{2}\right]=\{a+b\sqrt{2}:a,b\in\mathbb{Z}\}$ and $\mathbb{Z}\left[\sqrt{3}\right]=\{a+b\sqrt{3}:a,b\in\mathbb{Z}\}$ are subrings of $\mathbb{R}$. ...
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0answers
22 views

multiplicative inverse in factor ring

If I need to find the multiplicative inverse of an element in some $T[x]/(m)$ factor ring, do I need to solve a diophantine equation to get the solution? Let the element be $f$. Then $fu \equiv 1$ ...
1
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1answer
38 views

Is there a way to encode a ring into a group?

Is there a meaningful bijection between tne set of all rings and the set of all groups? Thanks.
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0answers
14 views

Neccessary and sufficient conditions to form a topological ring on $\Bbb{Z}$?

Let $B = \{ \{a + b f_i(n) : n\in \Bbb{Z}\} : a,(b\neq 0) \in \Bbb{Z}, f_i \in F \}$. Then what are necessary and sufficient conditions on the set of integer functions $F$ such that $B$ is a basis ...