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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Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
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Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b ...
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Equations in the semiring of f.g. modules

Let $R$ be a commutative ring. Then we may consider the semiring $G(R)$ of isomorphism classes of finitely generated $R$-modules with $+ = $ direct sum, $* = $ tensor product, $0 = $ zero module, $1 = ...
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Any simple proof of that localization of a UFD is a UFD without using Kaplansky condition?

Using Kaplansky condition, the proof of this statement is quite easy. Before knowing this condition, I tried to prove this statement by actually showing the definition of UFD holds on $S^{-1}A$ ($A$ ...
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Why is the completion of the ring of germs of smooth functions $\cong \mathbb{R}[|T|]$?

Let $C^{\infty}$ be the canonical commutative ring on the set $\{f: \mathbb{R} \rightarrow \mathbb{R} \mid f \text{ smooth}\}$. Let $\mathfrak{m}= \{ f \mid f(0)=0 \}$ a maximal ideal. Consider the ...
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$p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$

I need to show that $p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$ Here's what I've done: Please tell me if it's correct Over $\mathbb C,$ $x^4-2x^2-4\\=(x^2-1)^2-5\\=(x^2-1+\sqrt ...
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About a class of commutative rings that they have maximal ideals for any element non-inversible in $ZF\neg AC $: resolution

Let $\mathcal{N}{oetherian}\mathcal{C}\mathcal{R}{ng} \overset{def}{=} {\left\lbrace{ R \in \mathcal{C}\mathcal{R}{ng} \wedge R \,\text{is}\, \mathcal{N}{oetherian} }\right\rbrace}$. I define the ...
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A questions about Group Rings

Let's say $R:=\mathbb{Z}_p[C_{p^\infty}]$ be the group ring of a Prufer group over the field of integer module a prime $p$. We have $C_{p^\infty}=\langle u_1, u_2, ..., u_n, ... |\,\,\,\, ...
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Prime Ideals as Ring theoretic Ultrafilters

I am confused by the following statement in Awodey's Category Theory p. 35: Ring homomorphisms $A\to \mathbb Z$ into the initial ring $\mathbb Z$ play an analogous and equally important role [to ...
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A Lemma of Kaplansky

Source: Rings With a Polynomial Identity, Irving Kaplansky The Lemma: Suppose that $\mathbb{A}$ is an $\mathbb{F}$-algebra, where $\mathbb{F}$ is a field. Then, suppose that $\mathbb{A}$ ...
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A Commutator Identity in Rings

In a ring (or associative algebra), let the commutator $[A,B]$ be defined as $[A,B]=AB-BA$. I have asked earlier for a general formula for the expression $[x_1\cdots x_m,y_1\cdots y_m]$ in a group ...
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Counterexamples to the Artin-Rees Lemma

This well known Lemma about $I$-stable filtrations asserts: Lemma (Artin-Rees) Let $A$ be a Noetherian ring and $E$ a finitely generated $A$-module. Let $F$ be a submodule of $E$ and $\{E_i\}$ an ...
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$G \simeq R^{\times}$

What is known about the groups G for wich there exist a unitary ring R, such that $R^{\times} \simeq G$? I can easily prove that The only G cyclic with this property(Edit:and odd order) are those who ...
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Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
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Units and zero divisors in $\mathbb{Z_2[x]}/(x^2 + 1)$ and in general

The elements of $\mathbb{Z_2[x]}/(x^2 + 1)$ are polynomials in $\mathbb{Z_2[x]}$with degree $0$ or $1$. I.e. $\{\overline{0}, \overline{1}, \overline{x}, \overline{x+1}\}$ $\overline{0} = ...
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The integral closure of Bezout domains in arbitrary field extensions is Prüfer?

Let $R$ be a Bezout domain with quotient field $K$, $L$ an arbitrary extension field of $K$, and $\overline{R}$ the integral closure of $R$ in $L$. Is $\overline{R}$ a Prüfer domain? If the answer is ...
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Condition so that coefficients nilpotent implies power series nilpotent for non-noetherian ring $R$ with char$(R) = 0$

For a noetherian ring $R$ the following holds: $$\sum\limits_{i=0}^{\infty}a_iX^i \mbox{ nilpotent } \iff a_i \mbox{ nilpotent } \forall i, \mbox{ where } a_i \in R.$$ If $R$ is non-noetherian ...
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About quotient ring

I want to find the value $|\mathbb{Z}({\sqrt{2})/(3+\sqrt{2})}|,|\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}|$ and also the number of ideals of $\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}$. But still not ...
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When every matrix is a sum of nilpotent (idempotent or invertible) matrices ??

Let $R$ be a ring with non-zero identity. Consider the following three properties: Every $A \in M_n(R)$ is a sum of nilpotent matrices. Every $A \in M_n(R)$ is a sum of idempotent matrices. Every ...
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Weakening the assumption that an ideal is maximal

This question is from ChI, $\S{3}$ of Serge Lang's Algebraic Number Theory. Let $A$ be a commutative integral domain, integrally closed in its quotient field $K$, and let $E$ be a finite extension of ...
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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}) ...
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When powers of matrices are represented as a sum of integral matrices

There is given a ring $R$ and a subring $K$ with unit. We have a matrix $A$ of size $n$ over $R$. The characteristic of $R$ is $0$ or more than $n$. The statement is: If $A^m$ for any ...
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The action of a Galois group on a prime ideal in a Dedekind domain

Let $A$ be a commutative Dedekind domain and $K$ its field of fractions. Let $L/K$ be a finite Galois extension with Galois group $G$ and let $B$ be the integral closure of $A$ in $L$. If ...
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associated graded ring is the quotient of a free algebra by a homogeneous ideal

Let $A$ be a semilocal ring with Jacobson radical $m$ and let $I$ be an ideal of definition, i.e. an ideal such that $m^{\nu} \subset I \subset m$. Consider the associated graded ring of $A$, given by ...
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Hilbert symbol over a ring

Normally the Hilbert symbol over a field $\mathbb{F}$ is defined for $a,b\in\mathbb{F}^*$ as follows: $$ (a,b)=\begin{cases}1,&\text{ if }z^2=ax^2+by^2\text{ has a non-zero solution }(x,y,z)\in ...
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On complexes of projective modules

How can I prove the following statement? Let $\beta: B\rightarrow C$ be a quasi-isomorphism of complexes of $R$-modules. If $P$ is a complex of projective $R$-modules which is bounded below, then ...
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Applications of Govorov-Lazard Theorem?

The Govorov-Lazard Theorem states that a (right) module over an unital ring is flat iff it is a direct limit of finitely generated free (right) modules. I wonder if this theorem has interesting ...
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Extension of the theorem of Jacobson

Let $A$ be a ring. Let $E$ be the set of polynomials $\{X^n-X \in \mathbb{Z}[X]|n \in \mathbb{N}^*-\{1\}\}$. By the theorem of Jacobson, we know that if for each $a\in A$ there is an element of $E$ ...
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What's this called? $\mathbb{C}[d/dx]$

The 'ring of differential operators wrt x' ? Thx.
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Does the singleton reduction system $\{(x^2y,yx)\}$ lead to a normal form?

Suppose you have a singleton reduction system $\{(x^2y,yx)\}$. Does such a system lead to a normal form on the corresponding $k$-algebra $k\langle x,y\rangle$, where $k$ is a commutative, ...
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Dimension of a module vs codimension of its annihilator

For some reason I am just stuck on this question, although my intuition insists it's easy - I'd appreciate anyone telling me any obvious fact I'm overlooking here. Suppose we have an ...
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Projective modules over rings without unit

For rings with unit there are at least three ways to define a projective module: The universal property, i.e. $P$ is projective if for any epimorphism $M\to N$ and any morphism $P\to N$ there exists ...
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The number of local rings $R$ such that $R^\ast$ is cyclic of order $n$

For $n>0$, let $c_n$ be the number of local rings $R$ such that $R^\ast$ is cyclic of order $n$. Note that $c_1 =1$. (A local ring $R$ such that $R^\ast = \{1\}$ has precisely two elements. See Is ...
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Azumaya algebra and its subalgebras

I remind you that an Azumaya algebra $A$ is a central and separable algebra. Now, I know that if $A$ is an algebra over a skew-field or over a local ring then there exists a subalgebra $S$ of $A$ such ...
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Show $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to $\mathrm{E}(\mathbb{Z}_{m})\times\mathrm{E}(\mathbb{Z}_{n})$ if and only if $(m,n)=1$

Define $\mathrm{E}(\mathbb{Z}_{i})$ to be the group of invertible elements of the ring with unity $\mathbb{Z}_{i}$. Show that $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to ...
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Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
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Verifying three equivalent conditions for a ring to be Noetherian

Let $A$ a commutative ring with identity, then the following are equivalent Every set of ideals of $A$ has a maximal element Every ascending chain of ideals in $A$ is stacionary Every ideal in $A$ ...
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Maximal ideals of R[x]/(f(x))

I have been studying for my Qualifying Exam and came across the following problem: Let $R\subseteq T$ be integral domains and suppose that $a\in T$ satisfies a monic polynomial of degree $d$ with ...
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tensor products of Hopf algebras

Let $H_1,\cdots, H_n$ be Hopf algebras over the field $\mathbb{Z}_p$, $p$ prime. Then the tensor product $$ \bigotimes_{i=1}^nH_i $$ is still an algebra over $\mathbb{Z}_p$. Is $ \otimes_{i=1}^nH_i $ ...
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A field F can be extended to one in which all polynomials over F have a solution

Please check to see if my methods are correct. This question is a lot to absorb all at once. Let P be the set of non-constant polynomials over a field F and let X be the set of finite subsets of ...
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Question concerning a self-injective algebra and a faithful module

I'd like to know how corollary 2.11 of http://www.sciencedirect.com/science/article/pii/S002186930098726X# follows from theorem 2.10 from the same reference. 1) I know that $A$ being self-injective ...
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Application of the structure theorem for finitely generated modules over a PID

I've been trying to figure out the correct way to do this and I'm not sure I've got it quite right. $\mathbb{Z}^4/S$ where $S = \{(4, 12,20,8),(6,30,18,12),(4,16,16,8),(8,40,24,16) \}$ So, $M_0 = ...
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Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
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Uniqueness of the decomposition of an ideal

Let $ F $ be a non-empty subset of $ \{ 1,2,\dots,n\} $ and $ P_{F}=(\{x_{i}:i\in F\}) $. Let $ F_{1},F_{2},\dots,F_{m} $ be pair-wise distinct non-empty subsets of $ \{1,2,...,n\} $ and $$ ...
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Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
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Ring structure of a localization

let $R$ be a commutative noetherian ring and let $A$ be an $R$-algebra which is moreover a finitely generated $R$-module. Let $P$ be a prime ideal of $R$. How is the ring structure of the localization ...
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Idempotent semiring

Let $R$ be a semiring. For $a\in R$,we define $t_a(x)=x+a$ for all $x\in R$. Prove that $R$ is idempotent(with +) and $1$ has an infinite order if and only if for all $a,x,y\in R$, ...
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Multiplicative Inverse of Polynomials in Finite field

Find the multiplicative inverse of $x + 2$ in the field $\Bbb Z_5[x]/(x^2 + 2)$. I have done the following so far: \begin{align*} x^2+2 &= (x+2)(x+3) + 1\\ (x+2)(x+3) &\equiv -1 \pmod ...
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Showing function defined on $\text{Frac}(R)$ is a ring homomorphism

Let $f : R \to S$ be a ring homomorphism where $R, S$ are integral domains. I want to show that $\varphi : \text{Frac}(R) \to \text{Frac}(S)$ defined by $r/1 \mapsto f(r)/1$ is a ring homomorphism. ...
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Inverse limit of the following system.

Let $\mathcal{O}_L = \mathbb{Z}[\sqrt{-5}]$ be the number ring of $L=\mathbb{Q}[\sqrt{-5}]$, and $\mathfrak{p} =(2, 1+\sqrt{-5})$ a prime (and maximal ideal) in $\mathcal{O}_L$. What is the inverse ...