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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Simple composition factor

We know that if $e$ is a primitive idempotent in a semiperfect ring then $Re/Je$ is a simple left $R$-module, where $J$ is the Jacobson radical of $R$. Now, suppose that $e$ is an idempotent in an ...
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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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Rings With Bounded Index of Nilpotency are Dedekind Finite

Recently in an article by A. A. Klein I have seen this result: A ring $R$ with Bounded Index of Nilpotence is Dedekind Finite. Can anyone help me proving this result?
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For a given integer $n>1$ , for which type of rings $R$ is it true that $(xy-yx)^n=0 , \forall x,y \in R \implies R$ is commutative?

For a given integer $n>1$ , for which type of rings $R$ is it true that $(xy-yx)^n=0 , \forall x,y \in R \implies R$ is commutative ? (It is obvious indeed that if $R$ is an integral domain or a ...
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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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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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Field of polynomials mod n?

I have a few questions and i am looking for some clarification. 1) Is it correct that one can define a field $(Z_n, +, X)$ of integers mod $n$, where all the elements are integers $a$ such that ...
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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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Noetherian local ring with maximal ideal $M$

Let $R$ be a Noetherian local ring with maximal ideal $M$. If the ideal $M/M^2$ in $R/M^2$ is generated by $\{ a_1+M^2, \dots, a_n +M^2\}$, then the ideal $M$ is generated in $R$ by $\{ a_1, \dots , ...
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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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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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Prime ideals in $A$ and prime ideals in $S^{-1}A$

Let $A$ be a ring and $S$ be a multiplicative closed subset. Then there is a 1 to 1 correspondence between the prime ideals in $A$ (intersect $S$ is empty) and prime ideals in $S^{-1}A$. My question ...
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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 $q$ ...
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How many unique combinations of sets can we get?

Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists. Set $p$ a prime greater than 2, $\alpha = (p-1)/2$, and ...
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Does a ring map $f:R\to S$ induce a homomorphism $GL_n(R)\to GL_n(S)$?

Let $R$ and $S$ be commutative rings with $1$ and $f:R\to S$ a ring homomorphism. Does $f$ induce a group homomorphism $GL_n(R) \to GL_n(S)$? Progress I first consider the map $\bar{f}:M_n(R)\to ...
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Universal property of the Tensor Algebra

Let M be an A-module over a commutative ring A. For any A-algebra N and A-module homomorphism $\phi : M \rightarrow N$ there is a unique A-algebra homomorphism $\Phi : T(M) \rightarrow N$ (where T(M) ...
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Modules over PIDs

Let $M=\mathbb{Z}^4/N$ where $N$ is a subgroup of $\mathbb{Z}^4$ generated by $(1,0,-1,3)$ and $(2,4,8,-6)$. Recognize $M$ as a product of cyclic groups. Here I have to use the following Theorem: If ...
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Dixmier Conjecture

In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism. Which are some consequences of Dixmier conjecture ...
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Hartshorne, Exercise 3.18, Chapter 2

Let $B$ be a noetherian integral domain, let $A$ be a subring of $B$ such that $B$ is a finitely generated $A$ algebra. Assume that $A$ is also noetherian. Let $b$ be a non-zero element of $B$. How ...
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Basis of the ring $B=End_R(R^{(\mathbb N)})$

Let $B=End_R(R^{(\mathbb N)})$. Define $u,v \in B$ as $$u(e_{2_{i+1}})=0, u(e_{2_i})=e_i$$$$v(e_{2_{i+1}})=e_i,v(e_{2_i})=0$$ Prove that $\{u,v\}$ is a basis of $B$ as a $B$-module. I've already ...
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Unity of a subring of $\mathbb Z_{10}$

I've been told that $S=${$[0],[2],[4],[6],[8]$} is a subring of $\mathbb Z_{10}$ with unity $[6]$. How is it true though? $[2][6]=[12]=[2]$, $[4][6]=[24]=[4]$, and so on, isn't it? I realize I'm ...
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Surjective ring homomorphism from $M_n(R)$ to $M_n(R/I)$ where $R$ is a ring and $I$ is an ideal for R?

I'm looking for such a surjective homomorphism. I was thinking of starting from the canonical surjection from $R$ to $R/I$ and induce one but somehow I get stuck... Can you help me? Thank you very ...
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Unique maximal ideal in the ring of fraction

Let $R$ be a commutative ring with 1, and $P$ be a prime ideal in $R$. Let $D = R$ \ $P$. Show that $R_P := D^{-1}R$ has only one maximal ideal. Problem 2b in this link ...
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Maximality of the kernel of a quasifield

Setting A (left) quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is an abelian group. (As usual, we denote its identity element by $0$.) Each equation $x\cdot a = b$ with ...
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Every right principal ideal non-emptily intersects the center — what is that?

This is a follow-up to Do Lipschitz/Hurwitz quaternions satisfy the Ore condition? Jyrki Lahtonen answered the question in the positive by noticing that every right principal ideal in either ring has ...
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Factor rings of polynomial rings

Is there a unified explanation to the following phenomena? 1) $\mathbb{R} [X, Y] / (X^2 + Y^2 - 1) \mathbb{R} [X, Y]$ is not a UFD. 2) $\mathbb{C} [X, Y] / (X^2 + Y^2 - 1) \mathbb{C} [X, Y]$ is a ...
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Reference about Tensor Product

do you know some reference about tensor product of modules, with all elementary properties are proved ?? I want something a bit more explicit than "Commutative Algebra" of Atiyah and Macdonald. ...
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Question concerning the dedekind factoring of a principal ideal

In this paper the author gives the criterium for the factoring of a principal ideal of the ring of integers of a number field. My problem is how $[\mathcal{O}_K:\Bbb Z[\alpha]]$ is calculated. I think ...
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If $k$ is a field then $\text{End}_k(k^2)$ is simple

Let $k$ be a field. I have to show that $\text{End}_k(k^2)$ is simple. First of all, I don't see why this is true. For example, if $k=\mathbb{C}(x_1,x_2,\dots)$ then $\varphi\colon k^2\rightarrow ...
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Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
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Show that $31 | ord(\alpha)$ for a root of $f \in \mathbb{F}_{5}$

Let $f$ be an irreducible monical polynomial of in $\mathbb{F}_5[X]$ such that $\deg(f)=3$, and let $\alpha$ be a root in some field $\mathbb{F}_5^n$. Show that $31$ divides the order of $\alpha \in ...
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Rapid and easy question on ideals and ring

Let $R$ be the number ring related to a field $K$ of finite degree over $\mathbb Q$, i.e. $\mathbb Q\le K\le\mathbb C$ and $[K:\mathbb Q]=n$. Hence $R=\mathbb A\cap K$, where $\mathbb A$ is the ring ...
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Showing the existence of sub fields of a finite field

For each divisor $m$ of $n$, $GF(p^n)$ has a unique sub field of order $p^m$ . The proof of this theorem in Gallian goes like this : Suppose that $m$ divides $n$. Then, since : $(p^n-1) = ...
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Is this a fruitful enrichment of $R[X]$?

Let $R$ be a commutative ring. Then polynomial ring $R\left[X\right]$ can be looked at as an $R$-algebra free over a singleton. If $S$ is another $R$-algebra then for any element $s\in S$ there is a ...
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Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
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Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
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Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.

I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following. We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
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Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
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Example of a regular element with a commutative quotient

Whats an example of ring $A$ which is not commutative and contains an element $x\in Z(A)$ such that the left multiplication by $x$ is an injection and $x$ is not a unit and $A/(x)$ is commutative?
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Simple Cogenerator for the category of left $R$-modules

I want to prove that if the category of left $R$-modules has a simple cogenerator $M$, then $R$ is a simple ring. (Here, $R$ is an arbitrary ring with $1$.) My try is to take an ideal $I$ of $R$ ...
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Is every local ring the localization of some other ring?

One way of constructing a local ring is to start with any commutative ring, and localize all the elements outside of some maximal ideal (i.e., adjoining inverses to all those elements). But I'm ...
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What is $\operatorname{Ass}\operatorname{Ext}^i(M,N)$?

This is exercise 1.2.27 of Bruns-Herzog: Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $N$ an arbitrary $R$-module. Deduce that $\operatorname{Ass}(\operatorname{Hom}_R(M,N)) = ...
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Prime ideal is contraction of prime ideal iff it's saturated

Let $\varphi: A\to B$ be a commutative ring homomorphism and $P$ a prime ideal of $A$. The expansion of an ideal $I\subset A$ is the ideal generated by $\varphi(I)$ in $B$, and the contraction of an ...
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Graduations and filtrations for localizations

I'm trying to answer the following questions: Let $A$ be a (not necessarily commutative) $\mathbb{Z}$-graded ring and $S$ a multiplicative subset of $A$ such that $AS^{-1}$ exists. Is $AS^{-1}$ a ...
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Tensor product and projective dimension

Let $R$ be a local commutative Noetherian ring and be $M,N$ be finitely generated $R$ modules. Question$1$: If $\operatorname{pd}(M)$ and $\operatorname{pd}(M\otimes_{A} N)$ are finite ,then ...
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Finding a Prime Ideal in the Ring of $C^\infty$ Functions

Let $R$ be the ring of infinitely differentiable real-valued functions on $(-1, 1)$ under pointwise addition and multiplication, and let $$F(x) = \left\{ \begin{array}{lr} e^{-1/x^4} & ...
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Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...