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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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 ...
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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 = ...
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63 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 ...
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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 ...
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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 ...
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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 ...
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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 $F\cap R/P\neq (0)$. ($E$ ...
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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 ...
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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 ...
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104 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 ...
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137 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 ...
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116 views

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

If R is semiprime, right principally injective and satisfies ACC on right annihilators of elements, is it self-injective? I only know that it is right nonsingular.
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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 ...
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33 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
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45 views

A ring with a left cancellable element and a right identity always has an identity.

Let $R$ be a ring with $a, e \in R$ such that $a$ is not a left zero-divisor and $be=b, \forall b \in R$. Prove that $R$ has an identity. My attempt Let, $aeb = ab \Rightarrow aeb - ab = 0 ...
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43 views

Is the ring of germs of $C^\infty$ functions at $0$ Noetherian?

I'm considering the property of the ring $R:=C^\infty(\mathbb R)/I$, where $I$ is the ideal of all smooth functions that vanish at a neighborhood of $0$. I find that $R$ is a local ring of which the ...
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Let $R=k[[ X_1,X_2,X_3]]$ and $S=R/(X_1X_3,X_2X_3)$. Can one compute $\ell_S(S/(a_1^i,a_2^j))$?

Let $R=k[[ X_1,X_2,X_3]]$ and $S=R/(X_1X_3,X_2X_3)$. Let $x_i$ be the natural image of $X_i$ in $S$. Set $a_1=x_1+x_3$ and $a_2=x_2+x_3$. $a_1,a_2$ is a system of parameters of $S$. So ...
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24 views

prove n divides $[\mathbb{F}[\alpha]:\mathbb{Q}]$

$\mathbb{Q}<\mathbb{F}<\mathbb{C}$ - field extensions, such that $[\mathbb{F}:\mathbb{Q}]=m \in \mathbb{N}$ p is a prime number, $\alpha=p^{\frac{1}{n}}$ gcd(m,n)=1 prove n divides ...
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37 views

An ideal in a ring of polynomials and a field extension.

Let $K\subseteq L$ be fields and $I$ an ideal of $K[x_1,...,x_n]$. I want to show that $IL[x_1,...,x_n]\cap K[x_1,...,x_n] =I$. The inclusion $I \subseteq IL[x_1,...,x_n]\cap K[x_1,...,x_n]$ is ...
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17 views

Splitting field of a cubic polynomial understanding

The cubic polynomial $f(x) = x^3+px+q\in K[x]$ has 3 roots $a_1,a_2,a_3\in \mathbb C$ Hence, the splitting field extension $L=K(a_1,a_2,a_3)$ $\delta=(a_1-a_2)(a_1-a_3)(a_2-a_3)\in L$ since ...
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19 views

Condition for an integer prime to be a Gaussian prime

I have a basic question: To show that an integer prime $p$ is a Gaussian prime (i.e. a prime in the ring of Gaussian integers $\mathbb Z[i]$) if and only if the equation $x^2+y^2=p$ has no integer ...
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30 views

Diophantine linear Equation Gaussian Integers

We know that $ax+by=c$ with $gcd(a,b)=1$ could be solved over $\Bbb Z$. Supposing if $a,b,c\in\Bbb Z[i]$, is there an analogous framework to find $x,y\in\Bbb Z[i]$ (at least of minimum norms)?
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49 views

Determine whether a regular surjective map is finite

Consider the regular map between affine closed sets $f \colon \mathbb{A}^1 \rightarrow \mathcal{Z}(y^2-x^3) \subseteq \mathbb{A}^2$ given by $f(t) = (t^2,t^3)$. $f$ is obviously a dominant map. I ...
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28 views

Involutions on endomorphisms over division rings

Let $D$ be a division ring, and let $M $ be a free left $D$-module of finite rank. Assume that $x\mapsto x^*$ is an involution on the ring $\operatorname{End}_D(M)$ (which in this case means: ${}^*$ ...
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49 views

If $R$ is generated by idempotents, then $\text{Ann}(R)=0$?

Let $R$ be a ring (not necessarily commutative or unital) which is generated by idempotents. I'd like to know if $\text{Ann}(R)=0$ must holds. Here I use $\text{Ann}(R)$ to denote the set of all ...
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39 views

fundamental theorem of symmetric polynomials

Let $P_n(a,b,c)$ be a polynomial of variables $a,b,c$. By Newton's fundamental theorem of symmetric polynomials, there is a unique $P_n$ such that $$ x^n+y^n+z^n=P_n(x+y+z, x^2+y^2+z^2,x^3+y^3+z^3). ...
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Why Jacobson, but not the left (right) maximals individually?

When we are working with Path Algebras, it does not need very sophisticated tools to prove that for a finite, connected, acyclic quiver $Q$, the Jacobson Radical of $KQ$ is nothing but the arrow ...
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17 views

zero divisor graphs of a ring

If zero divisor graphs of two finite dimensional algebra are isomorphic does that imply that two algebra are isomorphic as a ring or a vector space?I would also like to know if graph theoretic ...
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29 views

The prime meadow of a meadow

Let $(R,(-)^{-1})$ be a meadow, i.e. $R$ is a commutative ring and $(-)^{-1}$ is a unary operation on the underlying set of $R$ satisfying $(x^{-1})^{-1} = x$ and $x \cdot x^{-1} \cdot x = x$ for all ...
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How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$ ?

How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$ ? I know that it is enough to determine $f([1]_{12})$ ; moreover $f([1]_{12}$ should be an idempotent element of $\mathbb ...
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Tidy way to represent XOR over the ring of $2^{32} - 1$

I was reading about a cipher called Speck, which defines a system of equations using Addition Mod $2^{32}$ ($\boxplus$), Bit Rotation, and XOR. If we pretend that the additions were taken over ...
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Module Theory: about the genesis of the concept

I'm studying the module theory. But, I really want to know the history of the genesis of the concept of R-module. On my years as a student, my Algebra teacher presenteed to me the concept as a simple ...
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43 views

How to show that this presheaf is a sheaf?

I am working with a presheaf of rings and I'm having problems to show that it is in fact a sheaf. Specifically: Let $A$ be a commutative ring with identity. Let $E(A)$ be it's ring of idempotents with ...
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26 views

Proof that ideal in Lie ring can be represented as sum of 2 Lie subrings

Let $K$ be a commutative ring and $m \ge 3$. Let $L(m,K)$ be a Lie subring of matrices with coefficients from ring $K$ that contains matrices with null traces, $L(m,K)=\{(a_{ij}) \in M_m(K) | ...
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show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent.

Show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent. The only idea that I come to mind is, we know $PR_P$ is the maximal ideal of $R_P$. Since $P$ is a prime ideal of ...
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68 views

Prime ideals in multivariate polynomial rings over $\mathbb R$ and in their quotients

1) Which are the prime ideals of $\mathbb{R}[X_1,\dots,X_n]$? 2) Which are the prime ideals of $\mathbb{R}[x,y]/\langle x^2+y^2-1\rangle$? About the question 2), I know that ...
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58 views

Chain of ideals in nilpotent algebra

Let $R$ be a nilpotent algebra ($R^n = \{0\}$ for some $n \ge 1$) and $A$ be a subalgebra of $R$. I want to show that exist a finite chain of subalgebras {$R_i$ | $i = 0, 1, ..., m $}, $m \ge 1$, ...
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50 views

Inverse of the product of fractional ideals

Let $R$ be a integral domain, $K$ its field of fractions and $\mathfrak M,\mathfrak N$ fractional ideals, i.e. non-zero finitely generated $R$-submodules of $K$. $$\mathfrak M^{-1}=\{x\in K: ...
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About non-zero divisors and related queries concerning endomorphism rings

Let $G$ be an abelian group. Then $\operatorname{End}(G)$, the set of all homomorphisms from $G$ to $G$, is a ring under addition defined as pointwise addition of functions and $.$ defined as ...
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Is the power of a finitely generated associative algebra still finitely generated?

Let $A$ be an associative finitely-generated algebra. I want to show that any power of $A$ is finitely-generated as well. Definition of power: $$A^k \stackrel{\text{def}}= \text{ { all finite ...
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61 views

How to solve this algebra problem?

Let $e$ be the idempotent element of the ring R. If $\langle e\rangle$ is the principal ideal generated with $e$, show that $R\simeq\langle e\rangle\times A(\{e\})$. I think $A$ s ring which contains ...
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62 views

What is the set of homomorphisms $\hom_\text{Ring} (\mathbb{Z}_{n},\mathbb{Z}_{m})$?

(For $A,B$ rings and $R$-modules, denote $\newcommand{\Hom}{\operatorname{Hom}}$ $\Hom_\text{Ring}(A,B)$ the ring of ring homomorphisms $A \to B$, contrasted with $\Hom_R(A,B)$ which is the R-module ...
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36 views

Using a Gauss sum to show that $p$ is of the form $x^2 + xy +3y^2$ if and only if $p \equiv 1, 3, 4, 5, 9 \pmod{11}$

Let $p \neq 11$ be an odd prime, and $\zeta$ an $11$th root of unity. Let $g$ be the Gauss sum $$g = \sum\limits_{i=1}^5 \zeta^{i^2} = \zeta + \zeta^4 + \zeta^9 + \zeta^5 + \zeta^3$$ We may ...
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$\mathbb{Z}[(1 + \sqrt{-7})/2]$ is euclidean

Show that $\mathbb{Z}\Bigl[\dfrac{1 + \sqrt{-7}}{2}\Bigr]$ is a Euclidean ring. Ok, there are some hints here but not a full proof. My attempt so far: Proof: Define $\omega := \dfrac{1 + ...
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66 views

Relatively prime polynomials over integrally closed domain are(?) relatively prime over the fraction field

I know that the following holds: Lemma. Let $R$ be an integrally closed domain and $K$ its field of fractions. Let $f \in R[X] \setminus R$ monic. Suppose that there exist $g,h \in K[X] \setminus ...
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Minimal Polynomial

Determine the minimal polynomial of $\frac{1}{\sqrt[5]{2}}+\frac{1}{11}$ over $\mathbb{Q}$. Put $x=\frac{1}{\sqrt[5]{2}}+\frac{1}{11}$. Put $x=\frac{1}{\sqrt[5]{2}}+\frac{1}{11}$. We need to ...
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31 views

Proof of Version of Krasner's Lemma

I need some help in constructing the proof to this version of Krasner's Lemma from Serre's Local fields text book: Let E/K be a finite Galois extension of a complete field K. Prolong the valuation ...
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32 views

Homomorphic images of $\mathbb{Z}[x]$

How to prove that any finite field is a quotient ring of $\mathbb{Z}[x]$ ? I am not sure whether this result is true or false. Any hint will be appreciated. Thanks in Advance.
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Is $r_1 \cdot f_1 + r_2 \cdot f_2 $ uniformly distributed?

Consider $f_1$ and $f_2$ are fixed polynomials, $r_1$ is a random linear polynomial, $r_2$ is a random polynomials, degree($r_2$)=degree($f_i$)=$d$. We define $f_i$ and $r_i$ over $R[x]$ where $R$ can ...
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To prove given $ r \cdot f_1+f_2\cdot (s+1)$ one who knows $f_2$ cannot find out what $f_1$ is

We define the polynomials $r,f_1,f_2,s\in R[x]$. Where $r$ is a random degree 1 polynomial and $s$ is a random polynomial such that: $\deg(s)=\deg(f_1)=\deg(f_2)$. Let $R$ be $\mathbb {Z}_q$ where $q$ ...