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

learn more… | top users | synonyms (1)

2
votes
0answers
81 views

A question about the consequence of Prime Avoidance.

I have found the following statement: Let $R$ be a Noetherian ring and $x$ is a non-zero divisor of $R$. Let $P$ be a prime ideal associated to $xR$. Then by Prime Avoidance there exists a non-zero ...
2
votes
0answers
72 views

A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
2
votes
0answers
75 views

Show that if $M$ is Noetherian then there $ n_{0} \in \mathbb{N}$ such that $n \geq n_{0}$, $0= \operatorname{Im}(f^{n}) \cap \ker(f^{n})$

Sean $M$ an $R$-module and $f: M \longrightarrow M$ an endomorphism of $M$. Show that if $M$ is Noetherian then there $ n_{0} \in \mathbb{N}$ such that $n \geq n_{0}$, $0= \operatorname{Im}(f^{n}) ...
2
votes
0answers
65 views

Functors that have a natural Isomorphism

Find different functors $T, S: Rng \rightarrow Rng$ both identity on objects IE: for each ring $R, T(R)=S(R)=R$, such that there is a natural isomorphism between T and S. I know that a natural ...
2
votes
0answers
106 views

In $ℤ/Nℤ$, which units are successors to zero divisors?

What are the units $x$ in $ℤ/Nℤ$ of the form $x = 1 + \overline{kd}$ for a divisor $d$ of $N$ and $k ∈ ℤ$, i.e. $$U_N[d] := \{x ∈ (ℤ/Nℤ)^×;\; ∃ k ∈ ℤ : x = 1 + \overline{kd}\} = \ker \big((ℤ/Nℤ)^× → ...
2
votes
0answers
91 views

Krull dimension of a module over the Weyl algebra

Let $A_{n}$ be the $n$th Weyl algebra over a field, with generators $x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n}$. Why is the Krull dimension of the rigth $A_{n}$-module $A_{n}/x_{1}A_{n}$ equal to ...
2
votes
0answers
46 views

Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$

I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
2
votes
0answers
65 views

Frobenius from Hurwitz's theorem

Can we deduce Frobenius theorem from Hurwitz's theorem on Normed division algebra? Frobenius theorem states that the only associative finite dimensional division algebras over the real numbers are R, ...
2
votes
0answers
73 views

A commutative ring with alternating and commutativity properties with infinite distinct elements

Is there any nontrivial commutative ring without multiplicative identity that satisfies alternating property ($x \cdot x = 0$ for all $x$ where $\cdot$ is multiplication operator and $x \cdot y \neq ...
2
votes
0answers
163 views

verifying if an ideal is prime

Let $R=\mathbb{Z}[\sqrt{-5}]$ and let $\mathfrak{i}=(2,1+\sqrt{-5})$ the ideal generated in $R$ by $2$ and $1+\sqrt{-5}$. I want to prove that $\mathfrak{i}$ is prime. So i considered the surjective ...
2
votes
0answers
58 views

Why $\text{top}h$ is an isomorphism?

I am reading the book Elements of representation theory of associative algebras I have a question about from Line -9 to Line -6 of page 29, the proof of Theorem 5.8. How to show that ...
2
votes
0answers
51 views

Left ideals of central simple algebra generated by symmetric element

Let $F$ be a field of characteristic $\neq 2$. Suppose $(A,\sigma)$ is central simple algebra over a field $F$ with an orthogonal involution. Assume $(A,\sigma)$ is non-split. How to show that every ...
2
votes
0answers
78 views

A necessary and sufficent condition for a ring to be a UFD

I came across the following question in Hungerford's Algebra: An integral domain $R$ is a UFD iff every non-zero prime ideal contains a nonzero prime principal ideal. The forward direction is ...
2
votes
0answers
88 views

Non-commutative integral extensions?

In Commutative algebra there is a notion of an integral extension: Let $P$ be a subring of $R$. Then $R$ is the integral extension of $P$ if each element of $R$ is a root of a monic polynomial with ...
2
votes
0answers
113 views

Multiplication structure for finite abelian rings of order $p^2$.

Consider an odd prime $p$ and a positive integer $a$ as close to 2 as possible that is not a quadratic residue $mod$ $p$. If we extend the ring $mod$ $p$ with the element $b = a^{\frac{1}{2}}$ then ...
2
votes
0answers
322 views

Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
2
votes
0answers
92 views

Projective dimension of simple module

Let $R$ be a commutative ring and $M$ a simple $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. Then it is known that $$ \mathrm{pdim}_{R}(M)=\mathrm{pdim}_{R_{\mathfrak{m}}}(M), $$ ...
2
votes
0answers
82 views

Writing out an isomorphism between two rings

I am having a hard time writing a bijective map between the two rings: $$ R = \dfrac{k[x,y,z,u,v]}{\left<(x-y)z+uv\right>} \cong \dfrac{k[x,y,z,u,v]}{\left<(x-y)z\right>} = S. $$ I ...
2
votes
0answers
36 views

$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...
2
votes
0answers
151 views

What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,…]$

What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,...]$? (this is the ring of polynomials over the reals with countably infinite many indeteminates). My attempt: I think taking the principal ...
2
votes
0answers
80 views

Degree 1 elements in a graded ring from a blow-up perspective

This may be an elementary question but I hope this question will benefit others as much as myself. Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$. Writing $Bl_{\mathcal{I}}(k^4)$ as $R = ...
2
votes
0answers
64 views

Simple $R$-module where $R$ is a semisimple ring. Possible small improvement of a proof.

Reading through the proof of the following theorem (in Introduction to Group Rings, by Milies and Sehgal) Let $L$ be a minimal left ideal of a semisimple ring $R$ and let $M$ be a simple ...
2
votes
0answers
73 views

Solving linear inequalities over rings

The concrete problem: for any given $N\ge 1$ I have a system of $2^N-1$ linear inequalities over $\mathbb{Z}_6^N$ which looks like this: for every nonempty $S\subseteq[N]$ there is some ...
2
votes
0answers
90 views

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent?

What does it mean for a subset of $\mathbb{C}[x_1,\dots,x_n]$ to be algebraically independent? Particularly I'd like to know the formulation thereof which concerns the kernel of a surjective ring ...
2
votes
0answers
67 views

Why is the $\mathbb{Z}$-span of a set of representations an ideal of the representation ring?

I am studying a proof of Brauer's theorem. The proof makes use of the following claim, which I haven't been able to convince myself of: Let $G$ be a finite group and let $R[G]$ be the representation ...
2
votes
0answers
115 views

Proving that a map is an isomorphism from an essential extension to itself.

I have a commutative ring $R$, and a prime ideal $P$ of $R$. I also have a module $E$ such that $R/P$ is a submodule of $E$ and every submodule $F\neq (0)$ of $E$ satisfies $F\cap R/P\neq (0)$. ($E$ ...
2
votes
0answers
142 views

Linearly independent rows in a square matrix

Suppose $m,n \in \mathbf{N}, m\le n$. Let $A$ be a matrix with $\mathbf{Q}$ linearly independent $b_{1},...,b_{m}$ in $\mathbf{Z}^{n}$. a) Show that there are $v_{1},...,v_{m} \in \mathbf{N}$ so ...
2
votes
0answers
181 views

A question about reduced torsion abelian groups

If a reduced torsion abelian group has no cyclic direct summands of order greater than 2, is it an elementary abelian 2-group? Background: I'm trying to classify the groups whose group rings have a ...
2
votes
0answers
105 views

Geometric understanding of principal/non-principal ideals

A number field $K$ with the $r$ embeddings into $\mathbb R$ and $2s$ pairs of conjugate embeddings into $\mathbb C$ can put into ring homomorphism with the product of rings $\mathbb R^r \times \mathbb ...
2
votes
0answers
139 views

Exterior algebras and radicals

So without giving excessive background, I was working on baby Spivak & got to the stuff about exterior algebras, & now I'm going through the section on tensor algebras in my copy of ...
2
votes
0answers
119 views

Why is this right principally injective ring a self-injective ring?

If $R$ is a semiprime ring, right principally injective and satisfies ACC on right annihilators of elements, is it self-injective? I only know that it is right nonsingular.
2
votes
0answers
173 views

What is it called when a subalgebra contains its centralizer?

In the question Math.SE #16716, Natalia asked about representing rings of matrices as centralizers of a matrix. This is an intriguing question, but had some clear problems as rings of matrices need ...
1
vote
0answers
30 views

How to prove that the ring of algebraic integers is a Bézout domain?

I was told that the ring of all algebraic integers (that is, the complex numbers which are roots of a monic polynomials with integral coefficients) is a Bézout domain. But I have no idea how to prove ...
1
vote
0answers
25 views

Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R.

Let $I, J$ be ideals of a ring $R$. Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R. Because $I,J$ are ideals of $R$, so $I,J$ both have $0$, thus $0+0=0\in I+J$. This shows ...
1
vote
0answers
19 views

Why are principal fractional ideal also fractional ideals?

I don't understand the following: Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ other than $\{0\}$, for which a $0\neq r\in R$ exists, so that ...
1
vote
0answers
19 views

If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.

Let $I$ be a left ideal in a semiprime ring $R$. If $I$ is a finitely generated semisimple left $R-$module, show that $I=Re$ for an idempotent $e \in I$.
1
vote
0answers
26 views

ring restriction of linear groups

Suppose that there is a group $G\subseteq\text{GL}(n,\mathbb{C})$ defined to be the group satisfying some equations on $\text{GL}(n,\mathbb{C})$(for example, given a nonsingular matrix $A$, ...
1
vote
0answers
24 views

Reduce $(\mathbb{Z}/n)^\times$ to $(\mathbb{Z}/p^n)^\times$

Please help me understand this problem: Given Problem: Using the ring isomorphism $\mathbb{Z}/rs \cong \mathbb{Z}/r \times \mathbb{Z}/s$, show that we have an isomorphism $(\mathbb{Z}/rs)^\times ...
1
vote
0answers
45 views

What is the name of this special subset of module over commutative ring with unity

Let $R$ ba a commutative ring with unity and let $M$ be a $R$-module. For each $m,n\in M$ consider free spaces $\bar{m}$ and $\bar{n}$ generated respectively by $m$ and $n,$ i.e ...
1
vote
0answers
31 views

How to show an ideal is principal

Is there a general procedure to check whether or not a prime ideal of the ring of integers $O_K$ is principal. In my case $K$ is a quadratic field, i.e $\mathbb{Q}(\sqrt {d})$, with $d$ square-free.
1
vote
0answers
23 views

Simple examples of fractional ideals

Let $K$ be the quotient field of an integral domain $R$. A fractional ideal $I$ is a subset of $K$ not $\{0\}$, for which a $0 \neq r \in R$ exists so that $r I \subseteq R$ is an ideal in $R$. ...
1
vote
0answers
23 views

Classification of algebras over GF(2)

I want to know something about algebras over GF(2). What are best structure theorem that a known at present? What I can read about this? Is there trivial excercise or it is not clear how make it now? ...
1
vote
0answers
35 views

Non-unique factorization in $\mathbb{Z}[\sqrt{-5}]$

I want to show that the decomposition into irreducible factors in the ring $$\mathbb{Z}[\sqrt{-5}] = \{a + b\sqrt{-5}|\space a, b \in \mathbb{Z}\}$$ is not unique, except for the order of factors ...
1
vote
0answers
66 views

Finding the left (or right) ideals of the ring of $n\times n$ matrices

Just give me a hint, since this is assessment! DO NOT TELL ME THE IDEAL I want to find the left (or right) ideals of the ring of $n\times n$ complex valued matrices. Now the definition is (for ...
1
vote
0answers
41 views

Krull's height theorem in the non-Noetherian case

Krull's height theorem says that if $R$ is a Noetherian ring and $I$ is a proper ideal generated by $n$ elements of $R$, then $\operatorname{ht} I\le n$. When $R$ is not Noetherian, this is not ...
1
vote
0answers
16 views

$x\in\mathcal{O}_K$ can be written as product of irreducible elements

Lemma: Every element $x\neq 0$ of $\mathcal{O}_K$ with ($K$ an arbitrary number field) can be written as a product of irreducible elements. Proof: We prove this lemma by complete induction on ...
1
vote
0answers
19 views

$x^{mn} -a$ is irreducible in F[x] iff $x^m -a$ and $x^n -a$ are irreducible.

Let F be any field, a is in F and (m,n)=1. Show that $x^{mn}-a$ is irreducible in F[x] iff $x^m -a$ and $x^n -a$ are irreducible in F[x]?
1
vote
0answers
39 views

Let $\mathbb{Z}[i]$ denote the Gaussian integers. The set of units of $\mathbb{Z}[i]$ is $\{\pm 1, \pm i\}.$

Let $\mathbb{Z}[i]$ denote the Gaussian integers. The set of units of $\mathbb{Z}[i]$ is $\{\pm 1, \pm i\}.$ A proof from: https://proofwiki.org/wiki/Units_of_Gaussian_Integers (Proof 2) Proof. ...
1
vote
0answers
51 views

Group Ring over commutative ring

$R$ is commutative semiprime ring, $(R,+)$ abelian group without torsion. Then $RG$ is semiprime. Proof by contradiction. If $x$ is nilpotent element in $RG$, then $x=r_1g_1+r_2g_2+...r_ng_n$. ...
1
vote
0answers
17 views

Does $U=U_1(\mathbb{Z}G)$ normalize $G$?

Let $G$ is an arbitrary group and and $U=U_1(\mathbb{Z}G)$ is the set of normalized units of $ZG$ i.e. $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. set of all ...