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Questions tagged [ring-theory]

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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Is the localization of a zero-dimensional ring a quotient?

If $R$ is any commutative zero-dimensional ring and $m$ is a maximal ideal, then is $R_m$ always naturally a quotient of $R$? In other words, is the natural map $R\to R_m$ always surjective? I was ...
Anon's user avatar
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4 votes
1 answer
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Attaching an element to a ring $R = \mathbb{Z}/(p^{k}\mathbb{Z})$, assuming it is not in $R$

Let $R = \mathbb{Z}/(p^{k}\mathbb{Z})$, where $p$ be any prime number, and $k > 1$ be any integer. Now let us consider an equation $x^r = p$ in $R$ and $\pi$ be the root of this equation, where $r \...
Afntu's user avatar
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Alternative solution to showing that $\langle x^2 +1, y\rangle$ is a maximal ideal and its possible generalization?

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg, and the following Notes: $\langle x^2 +1, y\rangle$ is maximal, pg.3 Question (5a) Background Notation 1: $\...
Seth's user avatar
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What do the words "descends" and "Induced" mean in the following quoted passage?

The following is taken from pg 4 section 6.1 of the following notes Background $\quad$ We start by observing that a ring homomorphism descends to quotient rings whenever the image of an ideal on the ...
Seth's user avatar
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3 votes
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Are there counterexamples of "dividing each other implies association" on a commutative ring but not integral domain? [duplicate]

I am reading about "Fraction on Commutative Ring". On the textbook a proposition states that Let $R$ be a domain and $a,b\in R$. If $a\mid b$ and $b\mid a$, then there exists a unit $u$ ...
Luost2r's user avatar
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What is wrong in this proof and where is it needed that $R$ is an integral doamin? [duplicate]

I wanted to prove the following theorem myself: Let $R$ be an integral domain and $p \in R$. If $(p)$ is a maximal ideal, Then $p$ is a prime element. My attempt: Since $(p)$ is maximal then there ...
Physor's user avatar
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Correspondence theorem and Quotient ring isomorphism

The following is taken from Abstract Algebra A Comprehensive Introduction by: Lawerence and Zorzitto. Background The ideal in a commutative ring $R$ generated by eleements $a_1,\dots, a_n$ is denoted ...
Seth's user avatar
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1 vote
1 answer
50 views

...if $\mathfrak{a}\subset\cup_{i=1}^{s}\mathfrak{p_i}$, then $\mathfrak{a_1}\subset \mathfrak{p_i}$ for some $i$

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg Background Definition 63 (Prime Ideal).: An ideal $\mathfrak{p}\neq R$ in a ring $R$ is a prime ideal if $rs\in ...
Seth's user avatar
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2 answers
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Relationship Between Subgroups of Abelian Groups & Ideals/Rings.

Clarification. I am currently reading from Dummit and Foote. Given $R$-module $M,$ we require $(1)$ $R$ is unital, and $(2)$ $1\cdot x=x$ for all $x\in M.$ When discussing rings $R,$ for the purposes ...
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When a quotient of $\mathbb Z_p[[x,y]]$ is a regular local ring

Consider the power series in two variables with p-adic coefficients. Namely consider $\mathbb Z_p[[x,y]]$. Moreover let $c\in\mathbb Z_p$ and construct the quotient: $$ W=\mathbb Z_p[[x,y]]/(xy-c) $$ ...
manifold's user avatar
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Is $k(w)=k$, where $w$ is algebraic over $k$, $f(w)=g(w)=0$, $f$ is even, $g$ is odd and $|deg{f}-\deg{g}|=1$?

Let $k$ be a field of characteristic zero, and $w \in \overline{k}$, an element in an algebraic closure of $k$, so $[k(w):k]=d \leq \infty$. Let $\alpha: k(t) \to k(t)$ be the involution defined by $...
user237522's user avatar
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Question about finding generators of the kernel for a substitution maps in rings.

The following is taken from Abstract Algebra A Comprehensive Introduction by: Lawerence and Zorzitto. Background Exercsie 6: Find generators for the kernel of each of the following substitution maps: ...
Seth's user avatar
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Conventional notation for gcd and principal ideal in the context of Bezout domain [duplicate]

Background Definition 1: Let $R$ be a commutative ring with identity, $c\in R$ and let $I$ be the set of all multiples of $c$ in $R$, that is, $I=\{rc\mid r\in R\}$. Set $I$ is an ideal and is ...
Seth's user avatar
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Questions about a proof of a theorem on prime ideals.

The following is taken from the text An Introduction to Grobner Bases by: Ralf Froberg Background Definition 63 (Prime Ideal).: An ideal $\mathfrak{p}\neq R$ in a ring $R$ is a prime ideal if $rs\in ...
Seth's user avatar
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1 vote
0 answers
40 views

Conceptual definition of the Auslander-Reiten translate

In homological algebra, we learn to differentiate between The conceptual definition. A computation, which is done by choosing efficient resolutions. The only definition I've seen of the Auslander-...
user135743's user avatar
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How to show normality is preserved under etale morphism?

Let $f:X\to Y$ be an etale morphism of Noetherian schemes, and $Y$ a normal scheme. How to show $X$ is a normal scheme? For $x\in X$, I only know that $\mathcal O_{Y,f(x)}\to \mathcal O_{X,x}$ is ...
Born to be proud's user avatar
1 vote
1 answer
26 views

Image of submodule to a quotient

Let $M,M'$ be $A$ modules. Let $N\le M$ be a submodule also $M' \le M$ is a submodule. Let $f:M\to \frac{M}{M'}$ be the natural $A$ module homomorphism. Claim, $f(N)= {(N+M')}/{M'}$ I wish to know if ...
Dinesh's user avatar
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Noncommutative analogue of ring of integers in number field

I'm studying Igor V. Nikolaev's paper "Untying knots in 4D and Wedderburn’s theorem". In the paper, he works with hyper-algebraic fields $\mathbb{K}$, i.e., fields with noncommutative ...
Ama's user avatar
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3 votes
1 answer
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Invariant Properties of Isomorphic Rings

I'm a second year maths student, looking at Rings and Modules questions for my exam. A property $P$ of rings is invariant under isomorphism if whenever $R$ is a ring with property $P$, and $S$ is a ...
CatsAndDogs's user avatar
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let $J$ be an ideal show that $rad(J)=J\cap rad(A)$ [closed]

Let $A$ be non-commutative banach algebra let $J$ be an ideal show that $rad(J)=J\cap rad(A)$ my attempt: let $x\in rad(J)$ then $x\in\bigcap M_i$ such that $M_i$ are maximal ideals of $J$ then $M_i=...
DARULE's user avatar
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3 answers
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$\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$?

I am exploring whether the following assertion holds true in integral domains: $\gcd(\text{multiples}(a,b))= \text{lcm}(a, b)$. Let us make this formal below. Consider two elements $a$ and $b$ in an ...
Martin Geller's user avatar
2 votes
0 answers
51 views

Maximal subgroups of multiplicative group of The Real Quaternions, $\textbf{H}(\mathbb{R})$.

The division ring $\textbf{H}(\mathbb{R})$ also known as the real quaternions, as it was described here, is an extention of the complex numbers. My main goal is to find or describe the maximal ...
Ash's user avatar
  • 492
2 votes
0 answers
53 views

A subfield $\mathbb{C}(u(t)) \subseteq \mathbb{C}(t)$ satisfying $\mathbb{C}(u(t),t^n)=\mathbb{C}(t)$

Let $u=u(t) \in \mathbb{C}[t]$, with $\deg_t(u)=d \geq 1$. Let $R=\mathbb{C}(u(t)) \subseteq \mathbb{C}(t)$ be the subfield generated by $u$. It is clear that $[\mathbb{C}(t):R]=d$. For every $n \geq ...
user237522's user avatar
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7 votes
1 answer
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A representation ring of all finite groups

Let $G_1$, $G_2$ be finite groups. Let $V_1$ and $V_2$ be finite-dimensional complex representations of $G_1$ and $G_2$, respectively. Then their tensor product naturally becomes a representation ...
Smiley1000's user avatar
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1 vote
0 answers
21 views

on the definition of prime ideals [duplicate]

An ideal $P$ of $R$ is said to be a prime ideal if and only if $P≠R$, and whenever $A$ and $B$ are ideals of $R$, then $AB⊆P$ implies $A⊆P$ or $B⊆P$. but what if $A⊆P$ and $B⊆P$? is it still a prime ...
Isaac 's user avatar
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2 votes
1 answer
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If $G$ is a group with each nontrivial element having an infinite order, does $\mathbb{Z}[\mathbb{G}] \cong 0$?

The definition of group ring in Wikipedia is Let 𝐺 be a group, written multiplicatively, and let 𝑅 be a ring. The group ring of 𝐺 over 𝑅, which we will denote by 𝑅[𝐺], or simply 𝑅𝐺 is the set ...
Jiahao Li's user avatar
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0 answers
31 views

Question about Noetherian rings [closed]

Is it true that: R is a Noetherian ring⇔Every countably generated prime ideal of R is finitely generated?
Patrick's user avatar
1 vote
0 answers
29 views

Wedderburn radical of a ring

There is a little information present in the literature about Wedderburn radical of a ring. Whenever I searched about Wedderburn radical of a ring, it shows about Wedderburn-Artin ring theory or ...
Muhammad Shanu's user avatar
2 votes
0 answers
28 views

Nilpotent subalgebras of $M_n(K)$

Let $A$ be a nilpotent subalgebra of $M_n(K)$ where $K$ is a field. Is it true that the index of $A$ is $\le n$? I know that if $A$ has a matrix with nilpotency index $n$ then it is true, since, its ...
eipi10's user avatar
  • 107
1 vote
1 answer
43 views

Does the set of all homomorphisms form a ring?

I know that the set of all homomorphisms $f:G\to G$, where $G$ is a group forms a semigroup with identity with respect to the composition operation. But does it form a ring with respect to addition ...
Isaac 's user avatar
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-1 votes
1 answer
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For certain field extensions $[L_1:F]<\infty$, $[L_2:F]<\infty$, $L_1 \simeq L_2$, when $[L_1:F]=[L_2:F]$?

Let $\mathbb{C} \subset F \subseteq L_1 \subseteq \bar{F}$, $\mathbb{C} \subset F \subseteq L_2 \subseteq \bar{F}$ be field extensions, with $[L_1:F]=n_1<\infty$ and $[L_2:F]=n_2<\infty$, $\bar{...
user237522's user avatar
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0 votes
1 answer
35 views

the ideal $〈6〉$ in the ring $(\mathbb Z,+,.)$

I am trying to solve this past exam question: In the ring $(\mathbb Z,+,.)$, the ideal $〈6〉$ is (a) maximal (b) prime (c) strongly prime (d) another answer. Which option is correct? The only theorem I ...
gbd's user avatar
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0 votes
1 answer
52 views

Trouble understanding an exercise needing to use $\Bbb{Q}[x]$ is algorithmic.

Background Exercise 17: This exercise shows that factorization in $\Bbb{Q}[x]$ is algorithmic. Let $f(x)\in \mathbb{Z}[x]$ be a polynomial degree $n$. If $f$ is reducible, it has a factor $g$ of ...
Seth's user avatar
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0 votes
1 answer
36 views

Meaning of the notation $\underline{x}$ for denoting coset.

The following is taken from Algebra: Notes from the underground by: Aluffi, Paolo Background Example 6.21 Consider the ring $R=\frac{\Bbb{C}[x,y]}{(y^2-x^3)}$. One can check that $R$ is an integral ...
Seth's user avatar
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-4 votes
0 answers
26 views

problem on prime ideal [closed]

Let (R,+,∗) be a ring and let (P,+,∗) be an ideal with P≠{0} (0 being the identity for +) and P≠R. Then (P,+,∗) is a Prime Ideal if for all a∗b∈P we have that a∈P or b∈P. Prove it.
Rahul Kumar's user avatar
0 votes
0 answers
37 views

Concerning some quadratic field extension $[L:\mathbb{C}(u,v)]=2$

In the following question I am actually asking about the answer to this MO question. First, I will ask it generally, then I will present the question and answer appearing in MO. A general question: ...
user237522's user avatar
  • 6,593
0 votes
1 answer
48 views

For a commutative ring $R$, is there an algebraic structure corresponding to $R/R^{\times}$, and does it have a meaning? [duplicate]

Question 1. For a commutative ring $R$, if we view $R$ as a commutative monoid, is there an algebraic structure corresponding to $R/R^{\times}$, and does it have a meaning? In detail, is $R/R^{\times}$...
with-forest's user avatar
  • 1,191
-1 votes
0 answers
28 views

Infinite direct product of rings (fields) [closed]

Let $R=\displaystyle{\prod_{n=1}^{\infty}}\mathbb{Z}_4$ be the infinite direct product of the ring integer modulo 4. I am looking for two ideals $I$ and $J$ of $R$ such that the ideals construct from $...
Tsafcko's user avatar
0 votes
1 answer
29 views

Root of an irreducible polynomial $g(x)$ over a finite field K = \frac{$\mathbb{F}$_p}{f(x)}

Question: Suppose $f(x)$ and $g(x)$ of degree n are irreducible polynomials over the field $\mathbb{F}_p$, where p a prime. Show that $g(x)$ has a root in $K = \mathbb{F}_p[x]/(f(x))$. How I ...
JelloHorse's user avatar
1 vote
0 answers
47 views

Confusion in proof of structure theorem for finitely generated modules over a PID

I'm working through a theorem in ring theory related to Principal Ideal Domains (PIDs) and finitely generated modules, and I've stumbled upon a statement about minimality that I cannot fully grasp. ...
Martin Geller's user avatar
0 votes
1 answer
53 views

What's the domain and range of a polynomial in a polynomial ring R[x] - definition

This might be a silly question but i can't find a straight answer on this.. What's the domain and range of a polynomial in a polynomial ring R[x] (where R is a ring)? is it all polynomials $p:R\to R$...
Ak2399's user avatar
  • 219
2 votes
0 answers
51 views

Different Notions of Maximal Ideals in Constructive Mathematics?

I was working on proving the following classic result for non-zero commutative rings in constructive logic: $I \subseteq R$ is a maximal ideal iff $R / I$ is a field. The definition of maximal ...
Léreau's user avatar
  • 3,113
1 vote
0 answers
32 views

Ideal generated by central element is central in a prime ring?

A ring $R$ is said to be prime if $xRy=0 \implies x=0$ or $y=0.$ Let $R$ be a prime ring and $x \in Z(R)$. Then the ideal generated by $x$ is central i. e. $\langle x \rangle \subseteq Z(R).$ I ...
MANI's user avatar
  • 1,958
2 votes
1 answer
48 views

Rings with unimodular matrices containing at least one unit

Let $R$ be a ring with the following property: "Every unimodular matrix (ie. with a determinant that is a unit in $R$) contains at least one entry that is a unit in $R$." It looks like a ...
Quiriacus's user avatar
0 votes
0 answers
11 views

Understanding the Relationship Between Closed Submodules and Pure Submodules of Finitely Generated Torsion-Free Modules Over an Integral Domain

Let $R$ be an integral domain, and let $M$ be a finitely generated and torsion-free $R$-module. In $(4.7)$ on the page 477 of the paper FINITISTIC DIMENSION AND A HOMOLOGICAL GENERALIZATION OF SEMI-...
Liang Chen's user avatar
1 vote
2 answers
80 views

Roots of $x^2+x+2$ over $\mathbb{Z}_3[X](i)$

As the polynomial $f(x)=x^2+x+2$ is irreducible, then $$ \mathbb{Z}_3[X]/\langle f(x) \rangle = \{p(x)+\langle f(x) \rangle \: : \: p(x) \in \mathbb{Z}_3[X]\} = \{ax+b+\langle f(x) \rangle \: : \: a,...
baristocrona's user avatar
1 vote
0 answers
58 views

$R = \mathbb R[X,Y]/(XY - 1)$ and $I$ be the ideal of $R$ generated by the image of the element $X - Y$ in $R$. Describe $R/I$

Let $R = \mathbb R[X,Y]/(XY - 1)$ ($\mathbb R$ is the set of real numbers) and I be the ideal of R generated by the image of the element X - Y in R. I want to find a way to describe R/I, i.e. find a ...
Jishnu's user avatar
  • 21
2 votes
2 answers
105 views

Number of polynomial functions in $\mathbb Z/2\mathbb Z[x_1, \, \ldots, \, x_n]$?

What is the number of polynomial functions in $\mathbb Z/2\mathbb Z[x_1, \, \ldots, \, x_n]$? (Here I define $p \sim q$ iff $p(x) = q(x)$ for all $x$.) What about the case if we allow permutations? ...
Markus Klyver's user avatar
0 votes
0 answers
28 views

If $J$ is a two-sided ideal of $k$-algebra $A\otimes_k B$, then $I=J\cap B$ is a two-sided ideal of $B$.

Let $A$ and $B$ be finite dimensional $k$-algebra, where $k$ is a field. If $J$ is an two sided ideal of $k$-algebra $A\otimes_k B$, consider $I=J\cap B$, I stuck with proving that I is an two sided ...
wwwwww's user avatar
  • 81
2 votes
1 answer
67 views

Is this a valid way of showing that $\mathbb C[x_1,\dots, x_n]/\langle x_1-a_1,\dots, x_n-a_n\rangle\cong \mathbb C$?

I know there are many ways of showing this isomorphism but I am wondering if this following one works. Define a ring homomorphism $\mathbb C[x_1,\dots, x_n]\rightarrow \mathbb C$ by $x_i\mapsto a_i$, ...
Chris's user avatar
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