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

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Let's cover-rings.

Let $R$ be a (commutative) ring with identity. A covering of R is a subset $\{r_1,...,r_n\}$ of elements of $R$ such that $R$ is generated by $r_1,...,r_n$. I think that if $\{r_1,...,r_n\}$ is a ...
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56 views

About the irreducibility of $p(x)=x^5+12ax^3+34bx+43c$ where $a,b,c$ are integers.

Consider the polynomial $p(x)=x^5+12ax^3+34bx+43c$ where $a,b,c$ are integers. Then $p(x)$ is irreducible over $\mathbb R$ if and only if $p(x)$ is reducible over $\mathbb C$. $p(x)$ is irreducible ...
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142 views

Understanding a proof of Wedderburn's little theorem

I am working on the proof of Wedderburn theorem and I have a problem to understand the part of it. I don't understand why $b_{1}^{-1}a_{1}=\lambda^{i}$ implies the last contradiction ...
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105 views

Computing ext over graded rings

This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that $A$ is a Koszul ring (for the definition of Koszul ring ...
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63 views

Why this ideal is generated by a finite set?

We know this known proposition of algebraic geometry: After that the author writes: Implicitly, he is saying this set $\{F^*\mid F\in I\}$ is necessarily finite, but I can't see why. Thanks in ...
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Lower bound for the degree of $h$ in $h= f^3 - g^2$ using Mason's Theorem

Let $h, f, g \in K[t]$, (let them also be non-constant) such that $ f^3 - g^2 \not = 0, f^3 - g^2 = h$. Show that $\deg(f) \le 2\deg(h) - 2$ and $\deg(g) \le 3\deg(h) - 3$. Alright, I know that this ...
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35 views

For certain prime ideal $\wp$, $(\wp^e)^c=\wp$?

Let $R\subset S$ be noncommutative rings, and let $\wp$ be a prime ideal in $R$, denote $\wp^e$ to be the extension ideal of $\wp$ in $S$, then my question is: Does $(\wp^e)^c:=\wp^e\cap R=\wp$ hold ...
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Commutativity of s-unital rings.

Theorem. Let $R$ be a left (resp. right) s-unital ring. If $R$ satisfy $(P_1)$ (resp. $(P_2)$). Then R is commutative(and conversely). $(P_1)$ $y^{s}[x,\, y]=\pm x^{p}[x^{m},y^{n}]^{r}y^{q}$ where ...
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Product of ideals closed set

Let $R$ be a topological ring (in fact $R$ is metrizable) and let $I,J$ denote ideals of $R$. Suppose also that they are closed with respect the topology of $R$. Is it always true that the product ...
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155 views

Equivalent conditions for commutativity in rings with 1

I have a problem with understanding the proof of theorem $3.1$. Could anyone help me with this? $(P_1)$ $y^{s}[x,\, y]=\pm x^{p}[x^{m},y^{n}]^{r}y^{q}$ where $m>1,r>0,n\geq0, s\geq0, p\geq0 , ...
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48 views

Can you put a ring structure on the group of units modulo $n$? What *other* rings are there?

Let $n=4$ Then with multiplication defined as: $$ \begin{bmatrix} 0 & 1 & 3\\ 1 & 3 & 0\\ 3 & 0 & 1 \end{bmatrix} $$ i.e. $a_{i,j} = i \oplus j$. Then combined with the ...
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118 views

A question on group rings

This looks like an elementary exercise on group rings (I heard it somewhere), nonetheless it seems to be non-trivial to me. Any references much appreciated. Suppose we are given an (infinite) group ...
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Can we always solve these linear algebra equations given rows of multiples?

We can find $n$ elements of multiplicative order $(n+1)$ modulo some large prime $p$, according to this question. Now I'm wondering if we can always perform linear algebra on the elements, as ...
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Image of a Euclidean domain

Considering Euclidean domains using a function $\varphi :E\setminus\lbrace 0\rbrace \rightarrow \mathbb{N}$ such that (i) $\varphi(a)\leq \varphi(ab)$ with $b\neq 0$. (ii) for all $a,b\in E$, $b\neq ...
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Is an invertible ideal in a semi-quasilocal ring a principal ideal?

Let $R$ be a semi-quasilocal ring and $I$ be an invertible ideal of $R$. Is $I$ a principal ideal of $R$?
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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 ...
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29 views

Shouldn't Gallian state $\Leftarrow$ for $a\ne0?$

Gallian text says in an integral domain $D,$ $a$ is a prime element $\iff(a)$ is a prime ideal of $D.$ Showing $\Rightarrow$ is easy. But I can see that $(0)$ is a prime ideal of $D$ (since $D/(0)$ ...
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28 views

(Quasi-)unmixedness and Completion

$(R,m)$ is a local ring and ${\widehat R}$ its $m-$adic completion. $R$ is said to be "unmixed" iff $$\forall p\in {\rm Ass} {\widehat R},\quad \dim {\widehat R}/p=\dim R$$ and is "quasi-unmixed" ...
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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)^× → ...
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46 views

What is the proper name of this class of rings?

There is a property of rings that seems to be quite natural, but I can't seem to find a short name for it. A commutative ring with unit $R$ has this property if and only if it has no divisors of $0$, ...
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50 views

Noetherian localizations and extra-condition implies Noetherian

I'm trying to solve this question but I'm stucked: If a ring $R$ satisfies the following two conditions: i) For every maximal ideal $M$ of $R$, if $S = R\setminus M$ then $S^{-1}R$ is ...
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66 views

How to show that semiring of sets is a semiring?

It is known that ring of sets(measure theory) is actually a ring by taking set operations, intersection and symmetric difference, as multiplication and addition. How can we do the same thing to ...
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Equality of two $k$-algebras

Let $f\in k[X_1,\ldots, X_n]$ and $1-fX_{n+1}\in k[X_1,\ldots, X_{n+1}]$. Moreover $X\subseteq k^n$ is a subset and $$I(X)=\{g\in k[X_1,\ldots, X_n]\,:\, g(x)=0\,\forall x\in X \}$$ is the ideal of ...
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normed division algebra

Can we prove that every division algebra over $R$ or $C$ is a normed division algebra? Or is there any example of division algebra in which it is not possible to define a norm? Definition of normed ...
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Triangular rings with direct sums

In the lam book ( a first course in non commutative rings), He is representing the triangular ring with direct sum!! I could not understand this part? How can we consider the triangular rings with ...
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50 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, ...
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Another problem with a proof in “Topics in Algebra” by Herstein- every element in R can be written as a finite product of prime numbers.

On pg. 146 of the second edition, it says let $a=bc$, where $a,b,c\in R$. $R$ is a Euclidean ring. If $d:R\to Z$, then if $b$ is a unit, $d(a)=d(c)$ [$d(a)=d(b)$ if $c$ is a unit]. If neither $b$ nor ...
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What is $\mathbb{C}[xy]/\langle x\rangle \subseteq \mathbb{C}[x,y]/\langle x \rangle$?

Consider the ring $\mathbb{C}[x,y]$, and consider $$R=\dfrac{\mathbb{C}[xy]}{\langle x\rangle } \subseteq \dfrac{\mathbb{C}[x,y]}{\langle x\rangle }\cong \mathbb{C}[y].$$ Is $R\cong ...
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Finitely generated ideal question.

Suppose $R$ is a ring, $I \subset R$ is an ideal, and $I = \langle S \rangle$ is finitely generated where $S \subset R$. Show that if $I$ and $J$ are finitely generated ideals of $R$, then so are $I ...
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Are there ways to get modulated values similar to cyclic and negacyclic convolutions?

The Wikipedia article on the Schönhage–Strassen algorithm states that there are methods that can get values modulo $a^n+1$ or $a^n-1$ for some value $a$. More specifically, it shows that the cyclic ...
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Ring of holomorphic functoins

Let $\ O_n =\{\text{all holomorphic functions around the origin in} \Bbb C^n\}$, I'm trying to prove the follwoing, if $f_i=z^2-w^{n_i},i=1,2$. then$$\frac{O_2}{f_1}\simeq\frac{ O_2}{f_2}$$if and only ...
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109 views

Showing that a Boolean algebra is a Boolean ring

I've proved that a Boolean ring is a Boolean algebra but I am having trouble with the converse. The operation for + is defined as the symmetric difference for elements $a$ and $b$ from the Boolean ...
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Isomorphisms betweenVerma modules over a semisimple Lie algebra

Fix a finite dimensional, semisimple Lie algebra $L$ and denote the Verma $L$-modules by $V(\lambda ')$ where $\lambda '$ are corresponding weights. Assume that there is an isomorphism between two ...
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Morphisms from the group variety of $n$-th roots

In Milne - Lectures on étale cohomology, example 6.10 i came across the following. We fix a variety $X$ and work in the category $Var/X$ of varieties over $X$ (so with fixed morphisms to $X$!) and ...
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How to compute $\gcd(a,b)$ if $N(a)=N(b)$?

Let $a,b$ be elements of an integral domain $R$. Let $N$ denote the norm. Let $x,y$ be other elements of the same integral domain $R$. I know that $\gcd(x,y)=\gcd(x,x-y)$ iff $N(x)>N(y)>0$. ...
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179 views

Why Does Induction Prove Multiplication is Commutative?

Andrew Boucher's General Arithmetic (GA2) is a weak sub-theory of second order Peano Axioms (PA2). GA has second order induction and a single successor axiom: $$\forall x \forall y \forall ...
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Why $\mathbb{C}[T_1, T_2]/(T_2^2 - T_1^3 - 1)$ is isomorphic to $\mathbb{C}[T, \sqrt{T^3+1}]$?

Is ring $\mathbb{C}[T_1, T_2]/(T_2^2 - T_1^3 - 1)$ isomorphic to $\mathbb{C}[T, \sqrt{T^3+1}]$? I know the first one is the ring of polynomial functions defined on curve $y^2 = x^3 + 1$.
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Is the group ring of a pro-finite group (semi)-hereditary?

It is well-known that a group ring of a finite group is semi-simple, and since profinite-groups are projective limits of finite groups, I am thinking per chance profinite groups still possess some ...
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Ring Theory and Induction

Andrew Boucher has developed a theory called General Arithmetic (GA): http://www.andrewboucher.com/papers/ga.pdf GA is a sub-theory of Peano Arithmetic (PA). If we add an induction schema (IND) to ...
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Is it true that when $ I$ has a term ordering on $R$ where $I$ is an ideal of $R$ then $I$ has a Grobner basis with respect to $\leq$

I don't think that this statement is true. Take for example, $I = (x^2 + y, x^2 y + 1)$. Clearly $I$ $\subseteq$ $k[x,y]$ and can have a term ordering yet $I$ is not a Grobner basis and since it's not ...
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Does an irreducible polynomial in K(t)[x] give an irreducible polynomial in K[t][x]

Let $K[t]$ be the ring of polynomials over a field $K$. Let $K(t)$ be its fraction field. Let $f$ be an irreducible polynomial in $K(t)[x]$. There exists an element $a\in K[t] $ such that $af$ is in ...
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159 views

Does every strongly $\pi$-regular ring have artinian prime factors?

A ring $R$ is called strongly $\pi$-regular if for every element $r \in R$ there exists an element $x \in R$ such that $r^{n+1}x = r^n$ for some positive integer $n$. Meanwhile, a ring $R$ is said to ...
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191 views

Question about the residue field of a localization

Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra which is an integral domain. Let $m$ be a maximal ideal of $A$. Does the proof that $A_m/mA_m$ (i.e. the residue field ...
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normalization of multiplicative subset of domain

I am stuck with this: Let $R$ be a domain with normalization $R' \subset K$. Show that for every multiplicative subset $S \subset R$, the normalization of $S^{-1}R$ equals $S^{-1}R'$. How do you ...
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143 views

All finite-dimensional simple modules are $1$-dimensional

Let $A$ be a (non-commutative) $k$-algebra, where $k$ is an algebraically closed, characteristic zero field. Let $M$ be a finite-dimensional simple $A$-module. If $A/\operatorname{ann}(M)$ is ...
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140 views

Intersection of closed sets in $\mathbb{Q}^2$

My question has connection with this question. Let $k>0$ be an integer without square factor. We consider the ring $\mathbb{Z}[\sqrt{k}]$. Let $N(a+b\sqrt{k}):= |a^2-b^2k|$ for $(a,b) \in ...
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Why do we have $\operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x\right>^i)\cong \operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x^2\right>^i)$?

I'm just curious but why is it that $$ \operatorname{Proj}\left(\oplus_{i=0}^\infty t^i \left< x\right>^i\right) $$ isomorphic to $$ \operatorname{Proj} \left(\oplus_{i=0}^\infty t^i \left< ...
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61 views

$S_k$ action on $A/I$

Let $S_2$ be a finite group of order $2$ and let $S_2$ act on $k[x,y]$ by interchanging $x$ and $y$, where $k=\overline{k}$. Then since $$ R = \left( \dfrac{k[x,y]}{(x+y)} \right)^{S_2} = ...
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Understanding $Bl_{\mathcal{I}}(k^4)/S_2$ where $\mathcal{I}$ is defined by $(x_1-x_2,x_3-x_4)$

Let $k_4=Spec(k[x_1, x_2, x_3,x_4])$ and $\mathcal{I}$ is the ideal sheaf defined by $(x_1-x_2,x_3-x_4)$. Then $$ Bl_{\mathcal{I}}(k^4) = Proj (\oplus_{i\geq 0} I^i t^i) $$ where ...
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Norm of the generators of a fractional ideal.

Let $\mathcal{O}_l=\mathbb{Z}[\frac{1+\sqrt{-l}}{2}]$ with $l$ a prime number congruent to 3 mod 4. Let $\mathfrak{a}$ be a non-principal fractional ideal of $\mathcal{O}_l$. My questions are: Why ...