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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1answer
187 views

Find all homomorphisms

Find all ring homomorphisms $\Phi$: $\mathbb{Z}_2 \rightarrow \mathbb{Z}_6$ and $\Phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2$.
2
votes
1answer
58 views

Properties of quasiregular elements in a matrix ring

I've been puzzling over one of the properties of quasiregular elements listed in the wikipedia article on the topic. An element of a ring $x$ is quasiregular (left, right) when $1-x$ has a ...
-1
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2answers
141 views

Ring of order $p^2$ and its characteristic

I suppose that this question might be very easy for some people. However, I have got problem to get it. Could anyone explain to me why the characteristic of a finite ring of order $p^2$ is $p$. I know ...
2
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0answers
60 views

If $\mathbb{C}[G]$ is Noetherian and $G$ has a representation on $V$, when must $V$ be finite-dimensional?

I know this is a bit vague, but please bare with me here. Let's assume that $G$ is a finitely-generated torsion group. I want to show that $G$ is a finite group if I add some conditions. I suspect ...
2
votes
1answer
98 views

Principal Ideal Groupring

Let $R[G]$ be a groupring (not necessarily commutative). Under which conditions on $R$ and $G$ is $R[G]$ a Principal Ideal Ring, respectively a Principal Ideal Domain (not commutative either)?
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2answers
116 views

is $I^2=I$ true?

Suppose $I$ is an ideal of a ring with $1$. I think that $II=I^2=I$ but I am stuck showing it. I can easily show that $I^2\subseteq I$, but I dont know how to show that $I\subseteq I^2$. So is it ...
3
votes
2answers
98 views

Difference between $R[c_1,c_2,\dots, c_n]$ and a finitely generated $R$-algebra.

What is the difference between $R[c_1,c_2,\dots, c_n]$ ($c_1, c_2,\dots, c_n\notin R$), where $R$ is a ring, and a finitely generated $R$-algebra? Is the difference that if $c_1, c_2,\dots, c_n$ ...
1
vote
1answer
207 views

$1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent [duplicate]

Let $(A, +, \cdot)$ be a ring with $1$. An element $a\in A$ is nilpotent if there exists $n\in \mathbb{N}$ so that $a^n=0$. Show that if $a$ is nilpotent then $1+a$ and $1-a$ are invertible.
2
votes
2answers
179 views

The Stone-Čech compactification of a space by the maximal ideals of the ring of bounded continuous functions from the space to $\mathbb{R}$

There is a claim that for any completely regular space, the maximal ideals of the ring of bounded continuous functions from $X$ to $\mathbb{R}$ forms the Stone-Čech compactification of $X$. How is the ...
1
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1answer
92 views

Noncommutative ring of order $np^2$

Could anyone help me to prove this theorem, please? Let $R_1$ be a ring of order $p^2$ which is the direct product of $C_p$ with itself and a minimal generating system for $R_1$ is $[(a,0),(0,a)]$, ...
2
votes
2answers
141 views

Prove $Z[\sqrt{d}] /(a + \sqrt{d}) \cong Z/nZ$

Let $a,d$ be integers with $d$ square free. Prove that $\mathbb{Z}[\sqrt{d}]/(a + \sqrt{d}) \cong \mathbb{Z}/n\mathbb{Z}$ where $n= |a^2- d|.$ I've tried attempting the problem by looking for a ...
2
votes
0answers
73 views

Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
2
votes
1answer
124 views

Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
4
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5answers
363 views

Isomorphism of polynomial rings in several variables

I have been struggling with the following problem: How can one prove that if there is an isomorphism between several variable polynomial rings over a field $K$, $ \varphi : K[X_1, \dots, X_n] \to ...
1
vote
1answer
401 views

Proof of Hilbert's Nullstellensatz, weak form.

The statement of Hilbert's Nullstellensatz, weak form, as given here is "Let $f_1,f_2,\dots,f_n$ be polynomials in $K[x_1,x_2,\dots,x_n]$, where $K$ is an algebraically closed field. Then $1=\sum{g_t ...
1
vote
1answer
51 views

A question on generators

Suppose $I$ is an ideal in a ring $R$ which is finitely generated. Suppose on the other hand that there is some (possibly other) set of generators $\{g_t\colon t\in T\}\subset I$ which also generates ...
0
votes
1answer
81 views

An Isomorphism of Rings

Let $R$ be the ring of Quaternions over $\Bbb{Z}_{(3\Bbb{Z})}$ ($\Bbb{Z}$ localized in $3\Bbb{Z}$). Is it true that $\frac{R}{J(R)}$ can be represented as $M_2(\Bbb{Z}_3)$ ? ($\Bbb{Z}_3$ is the ...
0
votes
1answer
122 views

Proof of Noether's Normalization theorem.

As stated here, Noether's Normalization Theorem states: Suppose that $R$ is a finitely generated integral domain over a field $K$. Then there exists an algebraically independent subset ...
49
votes
12answers
6k views

Do groups, rings and fields have practical applications in CS? If so, what are some?

This is ONE thing about my undergraduate studies in computer science that I haven't been able to 'link' in my real life (academic and professional). Almost everything I studied I've observed be ...
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1answer
99 views

If $P$ is a prime ideal of an integral domain $D$, then is $D$ equal to its localization at $P$?

I refer to this article on the localization of integral domains. Let $D$ be an integral domain, and $P$ a prime ideal of $D$. $$D_P=\{ab^{-1}\mid a\in D,b\notin P\}$$ Let us suppose $P\subset D$. ...
5
votes
2answers
147 views

characteristic prime or zero

Let $R$ be a ring with $1$ and without zero-divisors. I have to show that the characteristic of $R$ is a prime or zero. This is my attempt: This is equivalent to finding the kernel of the homomorphism ...
-1
votes
2answers
58 views

If a $k$-algebra is finitely generated, then does $k$ also have to be a finitely generated field?

Let $k$ be a field, and $A$ be a finitely generated $k$-algebra. Then does $k$ also have to be a finitely generated field? Motivation: Let $A$ be generated by $\{a_1,a_2,\dots,a_n\}$, and $k$ be ...
0
votes
0answers
46 views

in a finite ring, left inverse implies right inverse [duplicate]

Let $R$ be a finite ring with $1\neq 0$. Suppose there are $x,y\in R$ such that $xy=1$. I have to prove that this implies $yx=1$. I have read the question Left inverse implies right inverse in a ...
9
votes
2answers
206 views

Characterization of the field $\mathbb{Z}/2\mathbb{Z}$

Let $R \neq 0$ be a ring which may not be commutative and may not have an identity. Suppose $R$ satisfies the following conditions. 1) $a^2 = a$ for every element $a$ of $R$. 2) $ab \neq 0$ whenever ...
6
votes
3answers
519 views

Left Multiplication Ring Homomorphism

Assume we have a non-commutative unital ring $R$ and an element $r$ not in the center. Define a map $$\phi_r:R\rightarrow R$$ $$x\mapsto rx$$ Can this ever be a ring homomorphism? If it can be ...
1
vote
1answer
32 views

This ring with this metric can have just isosceles triangles

I'm trying to solve item (II) of this question: MY ATTEMPT Suppose $\rho(a,b)=2^{-m}$ and $\rho(a,c)=2^{-n}$ with $n\leq m$. Therefore, $a-b\in I^m\setminus I^{m+1}$ and $a-c\in I^n\setminus ...
0
votes
1answer
56 views

A homomorphism between two A-algebras.

A homomorphism between two algebras is described here. I want to describe a homomorphism $f:A[x_1,x_2,\dots,x_n]\to R$, where $R$ is an A-algebra. $A$ is a ring. Obviously, $A[x_1,x_2,\dots,x_n]$ is ...
0
votes
0answers
48 views

Ring structure on finite string of elements of a group

This is a reference request. Suppose $(G, \cdot)$ is a group and consider the structure on $G^{<\omega}$ where, for $\mathbf{p} = (p_1, \dots, p_n) \in G^{<\omega}$ and $\mathbf{q} = (q_1, ...
3
votes
3answers
138 views

isomorphic polynomial rings

I'm certain that this is a dumb question, but I'll ask anyway. I know that if $\theta : F \to K$ is a field isomorphism then we get an induced isomorphism $\varphi:F[x] \to K[x]$ such that $\varphi|F ...
0
votes
1answer
64 views

Is proving that a mapping maps every element of the domain and is surjective sufficient to prove that it is a ring automorphism?

The ring under consideration is $R[x_1,x_2,\dots, x_n]$. Shouldn't proving that a mapping $f:R[x_1,x_2,\dots, x_n]\to R[x_1,x_2,\dots, x_n]$ maps every element of the domain and is surjective ...
1
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1answer
219 views

Verifying whether a map is a polynomial ring automorphism

On pg.1, this article talks about an automorphism $f:R[x_{1},x_{2}]\to R[x_{1},x_{2}]$ ($R$ is a ring) defined by $$f(a)=a, \forall a\in R$$ $$f(x_{1})=x_{1}+x_{2}$$ $$f(x_{2})=x_{2}$$ An ...
1
vote
1answer
85 views

degree of remainder on division of multivariate polynomials

Let $f, g_1, \cdots, g_s \in \mathbb{R}[x_1,\cdots,x_n]$ and consider the division of $f$ by the $g_i$. Standard multivariate division algorithm will give $f = \sum_i a_i g_i + r$. I have been trying ...
8
votes
0answers
127 views

Let $A \subset B \subset C$ be subrings. If $C$ is finitely generated as an $A$-module, is $B$ finitely generated as an $A$-module? [duplicate]

Let $A \subset B \subset C$ be commutative rings. Suppose $C$ is a finitely generated $A$-module. Can we conclude that $B$ is a finitely generated $A$-module? Thoughts: Since $C$ is a finite ...
1
vote
2answers
126 views

Let $R$ be a ring with every element but $1$ having a left quasi-inverse. Then $R-\{1\}$ is a group under $a*b=a+b-ba$.

This question is related to exercise 1.51 from Rotman's "Introduction to the Theory of Groups". An element $a$ in a ring $R$ (with unit element $1$) has a left quasi-inverse if there exists an ...
6
votes
2answers
292 views

Polynomial rings — Inherited properties from coefficient ring

To avoid mixing up things, I wanted to collect properties a polynomial ring is inheriting from the coefficient ring and what property implies another. Let $R$ be a ring (what else do I need at which ...
0
votes
2answers
170 views

What does “defining multiplication in quotient rings” actually mean?

Say $R$ is a commutative ring and $I\in R$ is an ideal. Let us consider the quotient $R/I$. It is created by taking every element $a\in R$, and adding all the elements of $I$ to it. The elements of ...
21
votes
7answers
1k views

Does every unital ring contain all the integers?

Let us suppose there is a ring $R$ with the multiplicative identity $1$. We know that $1+r\in R$, where $r$ is any element of the ring $R$. Does this mean $1+1$ is also part of the ring, or does ...
0
votes
1answer
62 views

Help with a proof in Hungerford's book

I didn't understand a detail in this proof of this theorem: The definition of content: I didn't understand why $C(C(f)f_1C(g)g_1)\thickapprox C(f)C(g)C(f_1g_1)$ I need help only in this part. ...
2
votes
1answer
121 views

for prime ideals, the intersection of the squares is the square of the intersection?

Here is something that i proved and i would appreciate feedback on my proof: Proposition: Let $A$ be a commutative Noetherian ring and $p,q \in \operatorname{Spec}(A)$. Then $p^2 \cap q^2 = (p\cap ...
4
votes
1answer
322 views

If every nonzero element of $R$ is either a unit or a zero divisor then $R$ contains only finitely many ideals.

Let $R$ be a commutative ring with $1$, if $R$ contains only finitely many ideals, then every nonzero element of $R$ is either a unit or a zero divisor. I know it's true. How about the converse? i.e. ...
4
votes
1answer
121 views

Ring of integers of a degree $5$ extension

Consider the polynomial $P(X) = X^5 - X + 1 \in \mathbb{Q}[X]$, and let $x \in \mathbb{C}$ be a root of $P(X)$. Let $K = \mathbb{Q}(x)$. How can you prove that the ring of integers $\mathcal{O}_K$ is ...
0
votes
2answers
94 views

Additive group of a finite ring of square free order is cyclic

$R$ is a finite ring of square free order $n>1$. How to show by the basis theorem for finite Abelian groups that additive group of $R$ is cyclic group of order $n$?
8
votes
2answers
228 views

Sufficiently many idempotents and commutativity

It is a well-known result that if a ring $R$ satisfies $a^2=a$ for each $a\in R$, then $R$ must be commutative. See here for proof. I am wondering whether the same result holds for finite rings if we ...
3
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4answers
305 views

ring without zero-divisors

Suppose we have a ring (could be infinite) without zero-divisors. I have to prove that if $xy=1$ then also $yx=1$ for some $x$ and $y$ in the ring. I really need hints for this, because it seems I ...
2
votes
2answers
185 views

Finite field extension over $\mathbb F_2$

I don't see why $[L:K]=4$, where $L = \mathbb{F}_2(x,y) = \operatorname{Quot}(\mathbb{F}_2[x,y])$ and $K = \mathbb{F}_2(x^2,y^2) = \operatorname{Quot}(\mathbb{F}_2[x^2,y^2])$ Let $p(X) = X^2-x^2 ...
0
votes
1answer
137 views

Every element in a ring can be written as a product of non-units elements

I'm trying to understand a little detail in this proof: I didn't understand why in a ring we can always write an element as a products of non-units elements. I need help. Thanks in advance
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vote
2answers
115 views

characterization of Principal ideal rings

I'm thinking about these statements and I would like to know if I am right. A principal ideal ring $R$, by definition, is a ring whose ideals are principals, since $R$ is itself an ideal, $R=(x)$, ...
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vote
2answers
138 views

How to calculate the embedding dimension of this ring

Define $$R=\frac{k[x,y,z]_{(x,y,z)}}{(x^2z,y^2z)}.$$ How can I prove that $\operatorname{edim}R-\dim R\leq1$ (where edim means minimal number of generators of the maximal ideal) ? It seems to me that ...
4
votes
2answers
111 views

Are these polynomials irreducible in $\mathbb{Q}[x]$?

$$x^3+x^2+x+1$$ $$x^5+x^3+x^2+1$$ $$x^5+x^3+x+1$$ I've tried applying Eisenstein criterion but I can't figure it out. Thanks in advance.
0
votes
1answer
2k views

Ideals in the ring of Gaussian integers

What are the proper ideals, prime ideals, maximal ideals of $\mathbb{Z}[i]$, the ring of Gaussian integers. Check whether $(1+i)$ is prime or maximal ideal. Can someone help me please. I have ...