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'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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178 views

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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202 views

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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Question about the Chinese Remainder Theorem and the residue class ring ${\bf Z}/p\# {\bf Z}$

In a question that I asked on MO, Terence Tao observed that: The Chinese Remainder Theorem tells us that the residue class ring ${\bf Z}/p\# {\bf Z}$ is isomorphic (as a ring) to the product of ...
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38 views

Why is every non-zero element not a unit of this ring?

Consider the ring $\mathbb{Z}[\sqrt2]=a+b\sqrt2$ where $a,b\in\mathbb{Z}$. Now, if I am understanding the definition of units correctly, they are all the elements within the ring that have ...
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34 views

Euclidean domains and Fields

I've been wrtiting a chain of inclusions of algebraic structures as given at the end of this first paragraph on wikipedia: http://en.wikipedia.org/wiki/Euclidean_domain And I've been giving examples ...
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Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor.

Suppose a and b belong to a commutative ring and ab is a zero-divisor. Show that either a or b is a zero-divisor. My answer goes like this: If ab is a zero-divisor, then there exists a nonzero ...
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25 views

On units in subrings ( or ideal ) and quotient ring of ring with unity

Let $R$ be a finite ring with unity and $S$ be an ideal ( or subring ) , let $R^*$ be the group of units of $R$ and $S^*:=R^* \cap S$ , then does $|S^*|$ divide $R^*$ ? Moreover , if $S$ is an ideal ...
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19 views

Construct the rule in fields extensions

Let $f(x) = x^3+6x^2-12x+3$ Show that f(x) is irreducible over $\mathbb Q$ . Let $\theta$ be a real root of $f(x)$, which exists due to the intermediate value theorem. $\mathbb Q(\theta)$ consists of ...
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What are the ideals of $End(R)$?

Let $R$ be a ring (with unity if necessary) , then $End(R)$ i.e. the set of endomorphism of the ring $R$ (the set of all ring homomorphisms from $R$ to $R$ ) forms a ring under point-wise function ...
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Example of a maximal idea

Let $A$ be the set of bounded continuous functions from the set of real numbers to itself. Then $A$ is a ring under pointwise addition and multiplication. The set $I$ of all functions $f \in A$ ...
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Derived categories of filtered modules

For a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which ...
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Inclusion of subring in Ideal

Let $K$ be a commutative ring with mutpilicative identity and $m \ge 3$. Let $L(m,K)$ be a subring of Lie ring of matrices with coefficients from $K$ and traces = $0$: $ \{ (a_{ij}) \in M_m (K) | ...
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Symmetry of two-sided ideals

I was thinking about two-sided ideals and have some intuition-guided, soft questions regarding them. Since I don't have anyone to talk to about such subject matter, I thought I'd ask. Let a be an ...
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34 views

Quotients of successive powers of the augmentation ideal

For $H$ be any group, let $\mathbb{Z} H$ denote the integral group ring. Define $J(H)$ to be the augmentation ideal i.e $J$ is the kernel of the ring homomorphism $\mathbb{Z}H \to \mathbb{Z}$ sending ...
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Generalisation of chinese remainder theorem on ideals of ring without 1

Let $I_1,\dots,I_n$ be (two-sided) ideals of a ring $R$ (not necessarily with 1), which are pairwise co-maximal, i.e. $\forall i\ne j\in \mathbb{Z}_{[1,n]}$, $I_i+I_j=R$. Let $f:R\to R/I_1\times ...
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Localization of euclidean ring is euclidean?

I am trying to prove that a localization of a euclidean ring is euclidean, and the converse statement. I feel the basic definition of the norm is enough but I do not know how. Please note I am very ...
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A question about a system of congruent equations? Is there a unified proof by using ring theory?

In his book ``Topics in number theory, Volumes I and II''. William J. Leveque proved the following theorem(see page 34) Theorem A necessary and sufficient condition that the system of congruences ...
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Polynomials in the extension of a complete field

I need some help in answering this exercise from Serre's Local Fields textbook: Let K be a complete field, and let f(X) in K[X] be a separable irreducible polynomial of degree n. Let L/K be the ...
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Question on an $\mathbb{R} $-algebra

Define $[n] = \{1,\ldots, n\} $, where $n \in \mathbb{N}$ and define the $2^n$- dimensional $\mathbb{R}$-algebra $C_n$ as follows: Notation: Basis is $e_I$, where $I \subset \mathbb{N}$ and let ...
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Can we build infinite products in $k[[X]]$?

Let $P \in k[[X]]$, where $k[[X]]$ denotes the ring of formal power series over the field $k$. Is well defined $$\prod_{n\in \mathbb{N}}P$$ (i.e. the power to infinity of $P$?) By looking at the ...
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Proof of Gauss lemma

Gauss lemma Let $R$ be a UFD and $F$ its field of quotients. Let $f=\sum_{i=0}^n a_ix^i \in R[X]$ with $a_0 \neq 0$. If $p$ and $q$ are non zero, coprime elements in $R$, such that $\dfrac{p}{q} \in ...
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UFD and relatively prime elements

I've found the following statement at page 9 of Griffiths, Harris "Principles of Algebraic Geometry": Proposition. If $R$ is a UFD and $u$, $v \in R[t]$ are relatively prime, then there exist ...
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Non-Noetherian subring of F[X,Y]

I am trying to prove that, for a given field $F$, the subring $$R:=\{p(X,Y)=\sum c_{ij}X^iY^j \in F[X,Y] : c_{0j}=c_{j0}=0 \text{ whenever } j>0\}$$ of $F[X,Y]$ is not Noetherian. I think I ...
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Necessary and sufficient condition for a ring homomorphisms property

The question states: Let $R$ be a commutative ring with unity and let $A,B\subseteq R$ be two ideals, find a necessary and sufficient condition for $\mathrm{Hom}(R/A,R/B)=0$. Since ...
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prove that $(E_{p^n},*)$ is cyclic group

if $p \in$ $\mathbb{N}$ is a prime integer, how can i prove that $E_{p^n}$ the group of invertible elements of $\frac{\mathbb{Z}}{p^n\mathbb{Z}}$ is a cyclic group.
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cohomology ring of grassmannian

Let $G_k(\mathbb{R}^n)$ be the grassmannian consisting of all $k$-subspaces in $\mathbb{R}^n$. How to compute the cohomology ring $$H^*(G_k(\mathbb{R}^n);\mathbb{Z})$$ and what is the result?
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the numer of monic irreducible polynomials of degree $3$ in $\mathbb{F}_q$

I want to know how hany monic irreducible polynomials of degree $3$ there are in a field $\mathbb{F}_q$. The whole number of monic polynomials of degree three is $q^3$. Now I want to find out how ...
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Relative topological tensor product

I know that for a pair of topological rings $R$ and $R'$ (perhaps with some additional topological/functional-analytic hypothesis on them?) there exist two topological tensor produts, the injective ...
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Problems from Polynomial Rings. My attempt shown.

$1.$ Let $f(x) =a_mx^m+a_{m-1}x^{m-1}+ \cdots +a_0$ and $g(x) = b_nx^n+b_{n-1}x^{n-1}+ \cdots +b_0$ belong to $\mathbb Q[x]$ and suppose that $f \circ g \in \mathbb Z[x].$ Prove that $a_ib_j$ is an ...
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Automorphism of certain f.g. free modules

This is a quick question from Frohlich and Taylor's Algebraic Number Theory, II.4, p 94. Let $R$ be a Dedekind domain with quotient field $K$, $\mathfrak p$ is a non-zero prime ideal of $R$ and ...
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Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
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Ordered rings $R$ with $ab = ba$, $ab \leq ba$ or $ba \leq ab$ for all $a, b \in R$?

Ordered rings $R$ with $ab = ba$, $ab \leq ba$ or $ba \leq ab$ for all $a, b \in R$? Are there any good examples that are not also commutative rings? I can't seem to think of any.
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How “separable” (not in that sense) is a polynomial?

Since "separable" is used for different meaning in separable polynomial and separation of variable, I am having trouble searching for anything related to my question. So I hope someone can help with ...
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Proving that a field of characteristic $0$ is the field of fractions of a proper subring.

If $K$ is a field of characteristic $0$, $A$ is a subring of $K$ maximal subring of $K$ which doesn't contain $\frac{1}{2}$, and $F$ is the field of fractions of $K$, then I have proved that $K$ is ...
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Why is the norm of an ideal contained in that ideal?

Suppose $K$ is a number field and that $\mathcal{O}_K$ is the ring of integers of $K$. Now, let $I$ be an ideal in $\mathcal{O}_K$. I know that $N(I) \in I$, but I want to prove it. By definition, ...
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Rings with noncommutative addition

I was wondering if "rings" with noncommutative addition are studied at all? Of course, if a ring $R$ has a $1$, then for all $a, b\in R$, $a+a+b+b=(1+1)a+(1+1)b=(1+1)(a+b)=(a+b)+(a+b)=a+b+a+b$, from ...
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Correspondence theorem for rings.

Could someone provide a reference that includes a full and honest proof of the Correspondence Theorem for rings? Let $A$ be a multiplicative ring with identity and $I$ an ideal of $A$. There is a ...
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When are all (prime) ideals of an $R$-algebra, extensions of (prime) ideals of $R$?

Let $f:R\rightarrow R'$ be a homomorphism of commutative noetherian rings. When are all (prime) ideals of $R'$ extensions of (prime) ideals of $R$? Is it true for the case $R'$ is $R$-flat?
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Modules with finite injective dimension have $\omega_R$-resolutions

Let $(R,m,k)$ be a local Noetherian ring, $M$ a finitely generated $R$-module. How can I prove that $M$ has finite injective dimension if and only if it has a $\omega_R$-resolution? ($\omega_R$ is the ...
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On an invertible element for equivariant K-theory

Fix a positive integer $m$. Let $G = \lbrace h\in\mathbb C | h^m = 1\rbrace$ and $(X,\pi)$ the standard representation of $G$. Namely $X = \mathbb C$ and $\pi:G \to GL(X)$ is defined by $\pi(h)v=h v$ ...
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Question about Principal Ideals

I'm just learning basic ring theory and had a question about the definition of a principal ideal. For a commutative ring $R$ with unity, Fraleigh defines the principal ideal generated by $a\in R$ as ...
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Can someone explain to me this answer about subrings?

so I know how to prove that $\mathbb{Z}\left[\sqrt{2}\right]=\{a+b\sqrt{2}:a,b\in\mathbb{Z}\}$ and $\mathbb{Z}\left[\sqrt{3}\right]=\{a+b\sqrt{3}:a,b\in\mathbb{Z}\}$ are subrings of $\mathbb{R}$. ...
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Product Ring isomorphism question

$R$ is an arbitrary (non-unital) ring such that if $n\in \mathbb{Z}$ and $r\in R$, $nr=r+r+...+r$ ($n$ times) if $n\geq 0$ and $nr=(-r)+...+(-r)$ ($n$ times) if $n<0$, where $r+(-r)=0$. Now let ...
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Normal ring and unmixed ideals

Let $R$ be a commutative Gorenstein local ring , $I$ an ideal of $R$ . If $R/I$ is normal ring , then for any $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?