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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2
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1answer
119 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 ...
0
votes
1answer
26 views

Notation Question about Rings

If $S = \langle2\rangle$ is the ideal generated by $2$ in $\mathbb{Z}$, what does $S[x]$ represent?
3
votes
2answers
70 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 ...
0
votes
0answers
38 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 ...
4
votes
1answer
142 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 $K[x^4,x^3y,xy^3,y^4]$ are considered the same. Is it true? Why? I'm a beginner so please answer in detail.
1
vote
1answer
42 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 ...
0
votes
4answers
82 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 ...
1
vote
1answer
20 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$? ...
-1
votes
1answer
58 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 ...
1
vote
1answer
54 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. ...
1
vote
3answers
31 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 ...
3
votes
0answers
78 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 ...
1
vote
1answer
89 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) = ...
1
vote
2answers
32 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 ...
1
vote
1answer
152 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 ...
6
votes
2answers
116 views

Prove $M$ is a Maximal Ideal in $\Bbb Z\times \Bbb Z$

A problem from introduction to abstract algebra by Hungerford. It asks: If $p$ is a prime integer, prove that $M$ is a maximal ideal in $\mathbb Z \times \mathbb Z$, where $M =\{(pa,b)\mid a,b\in ...
2
votes
2answers
50 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 ...
0
votes
2answers
56 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 $R$?
2
votes
0answers
36 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 $\pi(h)v=h v$ ...
9
votes
1answer
246 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 ...
1
vote
3answers
74 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 ...
1
vote
2answers
110 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 ...
0
votes
2answers
77 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 ...
0
votes
1answer
44 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 ...
2
votes
2answers
311 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 only finitely many prime ideals ...
0
votes
1answer
34 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 ...
1
vote
1answer
74 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. ...
-1
votes
2answers
69 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 ...
1
vote
2answers
162 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 ...
0
votes
2answers
164 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 ...
9
votes
1answer
169 views

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. $f$ ...
0
votes
1answer
95 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) ...
2
votes
0answers
64 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 ...
0
votes
3answers
59 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 ...
0
votes
2answers
158 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 ...
1
vote
0answers
66 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 ...
2
votes
0answers
60 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 $\mathbb{R}$. ...
3
votes
0answers
87 views

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

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 ...
0
votes
0answers
108 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
vote
1answer
121 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 ?
3
votes
1answer
144 views

Exercise 17.2 of Matsumura about CM rings

I have two questions about this exercise of Matsumura: Question 1: Why $y^3$ is $R/(x^3)$ regular? Question 2: I hardly (in 20 lines) can prove that $k[x^4,x^3y,xy^3,y^4]$ is not CM. Is there a ...
1
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0answers
20 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 ...
2
votes
3answers
463 views

Nilpotent elements in $\mathbb{Z}_n$

I'm trying to show that $\mathbb{Z}_n$ has a nonzero nilpotent element if and only if $n$ is divisible by the square of some prime. I have figured out the proof of showing that if $n$ is divisible by ...
2
votes
0answers
77 views

Product Ring isomorphism question

$R$ is an arbitrary (non-unital) ring such that if $n\in \mathbb{Z}$ and $r\in R$, $nr=r+r+...+r$ ($n$ times) if $n\geq 0$ and $nr=(-r)+...+(-r)$ ($n$ times) if $n<0$, where $r+(-r)=0$. Now let ...
1
vote
0answers
30 views

All topology pairs $(X,Y)$ such that $f: X \to Y$ is continuous.

Given an arbitrary function, or more specifically if you want let $R$ be a ring and let $X = S \times S; Y = R; S \subset R$ and $f(a,b) = a - b$, is there something interesting about all the topology ...
1
vote
1answer
32 views

Show $I=p\mathbb{Z}$ for prime $p$.

Let $I\subset\mathbb{Z}$ be an ideal such that $I\neq \mathbb{Z}$ and if $I\subset J\subset\mathbb{Z}$ then $I=J$ or $J=\mathbb{Z}$. Show that $I=p\mathbb{Z}$ for some prime $p$. Attempt: We know ...
1
vote
2answers
67 views

Is there a way to remove elements from $\Bbb{Z}$ and create a related ring structure?

For example. Everyone hates $3$, so let's remove it all-together from $\Bbb{Z}$: Let $\Bbb{Z}' = \Bbb{Z} - 3\Bbb{Z}$. Then is there a way to keep it a ring, i.e. $\Bbb{Z}'$ forms a ring by ...
1
vote
1answer
45 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
90 views

Completion of integral domain

Let $A$ be an integral domain with the $I$-adic filtration. Let $B$ be the fraction field of $A$. My question is the following: Is the fraction field of the completion of $A$ the same as the ...
0
votes
1answer
59 views

Set of non-units in a ring

Let $R$ be a ring with identity. Let ${\rm rad}\: R$ be the radical of $R$, ie the intersection $\bigcap L$ over all maximal left ideals $L$ in $R$. Let $S$ be the set of all non-units in $R$ ...