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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Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
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56 views

About a class of commutative rings that they have maximal ideals for any element non-inversible in $ZF\neg AC $

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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19 views

Neccessary and sufficient conditions to form a topological ring on $\Bbb{Z}$?

Let $B = \{ \{a + b f_i(n) : n\in \Bbb{Z}\} : a,(b\neq 0) \in \Bbb{Z}, f_i \in F \}$. Then what are necessary and sufficient conditions on the set of integer functions $F$ such that $B$ is a basis ...
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28 views

All topology pairs $(X,Y)$ such that $f: X \to Y$ is continuous.

Given an arbitrary function, or more specifically if you want let $R$ be a ring and let $X = S \times S; Y = R; S \subset R$ and $f(a,b) = a - b$, is there something interesting about all the topology ...
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14 views

Creating Polynomial

By relative prime factor theorem $$R = (Zm,+,.)$$ where R is the ring structure the input is $e_0 = 0$ and $e_1=1$ output is $$S_0 = { k : \gcd(m,k)>1 }$$ $$S_1 = { k : \gcd(m,k) = 1}$$ Now ...
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87 views

What are the consequences of presentation of an algebra by generators and relations?

Let $A$ be a finite dimensional associative $K$-algebra, where $K$ is a field. I wonder how the presentation of $ A $ by generators and relations helps in the study of structure of the algebra ...
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52 views

Proof for maximal ideals in $\mathbb{Z}[x]$

I have been trying to prove the following theorem: Every maximal ideal in $\mathbb{Z}[x]$ has the form $(p, f(x))$ where p is prime integer and f is primitive integer polynomial that is irreducible ...
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93 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
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55 views

Class Group of Ring of Integers of $\mathbb{Q}[\sqrt{-57}]$

Let $R$ denote the ring of integers of the imaginary quadratic number field $\mathbb{Q}[\sqrt{-57}]$. I must find the ideal class group $\mathcal{C}$. Using the Minkowski Bound, I know that I need ...
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37 views

Generators of a monomial Ideal

I am trying to determine the set that generates a monomial ideal. Namely, the ideal $(xy,yz,xz)^3$. I know it will have terms $x^3y^3$, $z^3y^3$, $x^3z^3$. For the other terms that generate, I do ...
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68 views

Proposition 17, p. 68 of Lang's Algebraic Number Theory

I'm stuck on a detail of the following propositon: Let $K, E$ be linearly disjoint number fields with degrees $n$ and $m$ over $\mathbb{Q}$ whose discriminants (over $\mathbb{Q}$) are relatively ...
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28 views

Grading and commutators

If $R$ is a unital associtaitve commutative ring, then can any $R$-algebra $A$ may be filtered as $A_0:=A$ and $A_{i+1}:=(A_i,A_i)$ where $(-,-)$ is the commutator of $A$ with respect to $A$'s ...
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30 views

How to show that the dimension of a quotient space in the field of polynomials is not finite?

I have to show that if I have a quotient of the form $\mathbb{K}[x_1,x_2,\dots,x_n]/\langle f_1,f_2,\dots,f_s\rangle$, $\operatorname{char}(\mathbb{K})\not=0$, and on which the class of $[x_i^l]$ is ...
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50 views

Are Ideals and Varieties Inclusion Reversing?

Let $S_1$, $S_2$ be sets or varieties (I don't think it matters, does it?). Then if $S_1 \subset S_2$, is it always the case that $I(S_2) \subset I(S_1)$ (where I is an ideal)? Also, is it always the ...
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Why can an ideal generated by a subset be written in this form?

I have a subset $F \subset R$ that generates an ideal $(F)$. Apparently this can be written in the form $$(F)=\{a_1f_1b_1+...+a_kf_kb_k|k \ge 0, f_i \in F, a_i,b_i\in R\}$$ or if $R$ is commutative ...
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17 views

Examples of d-extensions in realisation of $\operatorname{Ext}^d$

If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at ...
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54 views

Are these cubic rings the same?

Consider the pure cubic field $K=\mathbb{Q}(\sqrt[3]{10})$ then as $10\equiv 1 \pmod 9$ then the integral basis for $K$ is of the form $\{1,\sqrt[3]{10},\frac{1+\sqrt[3]{10}+\sqrt[3]{10^2}}{3}\}$. And ...
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42 views

How can we show that $\operatorname{Tor}_1^A(\hat{A},G_f/G)=0$?

This question is quite closely related to my last question: Is it always true that $\hat{A} \otimes_A A_f/A \cong \hat{A}_f/ \hat{A}$? Let $A$ a commutative ring, $f \in A$ a non zerodivisor. Let ...
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62 views

Noetherian Ring and Homomorphic Image

Prove that, if $R$ is Noetherian, then so is each homomorphic image of $R$. I know that by the Fundamental Homomorphism Theorem this is the same as showing that if $R$ is Noetherian, then so is ...
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35 views

Finding all subrings with identity of $\mathbb Z_{16}$

I have to solve following question but I cannot derive the solution. Find all subrings with the identity of the ring $\mathbb{Z}_{16}$. I'll appreciate if you could help me to solve this ...
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42 views

prove that the ideal $I=(x,xyx,\dotsc,xy^nx,\dotsc)$ cannot be generated by a finite number of polynimials.

The ring is $K\langle x,y \rangle$ So we have to assume that exist a finite number. Let $f_1,\dotsc,f_m$ a finite number of polynomials. How we can arrive at a contradiction?
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86 views

Properties of $\text{gcd}$ in Integral domain

Let $R$ be an integral domain in which every pair of elements of $R$ has $\text{gcd}.$ Suppose $r\in R$ is irreducible and $r \nmid a, r \nmid b \ (a,b \in R).\ $ I have established that $a= ...
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100 views

Relation between finite stable rank and IBN (invariant basis number)

For R is any ring has an identity, we known if R has stable rank one, then R is "weakly finite" (or "stably finite," all matrix rings over R are Dedekind finite) and this implies R has IBN . But ...
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29 views

integral basis for an arbitrary cubic Galois field

I wonder where I can find some information (possible a book) about finding an integral basis for cubic Galois fields? I know that for pure cubic fields there exists a simple criterion according to ...
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40 views

Integral Domain and PID Proof

Prove that, in a domain, $(a)=(b)$ iff $a = bu$ for some unit $u$. By $(a)=(b)$, it also means that $a\mid b$ and $b\mid a$ so we can write them as $a=bu$ and $b=av$ for some $u, v \in R$ where $u$ ...
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56 views

Isomorphism and crossed product

When I consider isomorphism of matrix rings and crossed product, I want to know the answer of the following question. Thanks. Let $R$ be a ring and let $$T(R)=\left\{\begin{pmatrix} a & b & ...
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representation type of PI rings

A ring $R$ is said to satisfy a polynomial identity (PI for short) if there exists a polynomial $f(x_1, \ldots, x_n) \in \mathbb{Z} \langle X_1, \ldots, X_n \rangle$ such that $f(r_1, \ldots, r_n)=0$ ...
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44 views

Faithfully flat checkable on finitely generated modules

A left $R$-module $_RM$ is said to be faithfully flat if it is flat and, for any $N_R$, $N \otimes_R M = 0$ implies $N = 0$. I would like to show that $M$ is faithfully flat if it is flat and, for any ...
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48 views

dimension of tensor products over a submodule

Let $k$ be a field, $A$ be a finite dimensional $k$-algebra (say of dimension $n_A$) and let $B\subset A$ be a sub-algebra. What can be said about the $k$-dimension of $A \otimes_B A$ ? The easiest ...
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43 views

Enveloping Algebra equal to algebra

Let $R$ be a unital associative ring, $A$ be an associative $R$-algebra of finite dimension, and $A^e$ its enveloping algebra. What are the requirements on $A$, so that $A^e \cong A$ (as ...
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137 views

LCM generators for the intersection of non-principal ideals in a Noetherian UFD

I am working with some non-principal ideals $I=\langle a,b\rangle$, $J=\langle c,d\rangle$ in a nicely behaved Noetherian UFD (the Laurent polynomial ring in finitely many commuting variables with ...
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39 views

kernel of maps associated to the root of an irreducible polynomial

Let $m(\mu)$ be an irreducible polynomial of degree $d$ over $\mathbb{F}_2$, $F_{2^d} = \mathbb{F}_2[x]/(m(\mu))$ by a field extension given by that polynomial and let $d: \mathbb{F}_2[x] \to ...
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32 views

Monoid Ring of a Commutative Cancellative Ordered Monoid

Suppose $M$ is a commutative cancellative monoid with $0$ as the identity and $+$ as the operation, and $M$ is equipped with an order $\preceq$ defined by $$ m \preceq m' \text{ if and only if there ...
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142 views

Unique factorization domain and principal ideals .

If R was a unique factorization domain, can we deduce that for a nonzero element d in R, d has a finite number of divisors? I need this in solving this question " If R is a unique factorization ...
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55 views

Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...
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Hochschild homology with trivial coefficients: how to make $K$ an $M_n(K)$-module

Let $R$ be a ring, $A$ an associative $R$-algebra, and $M$ an $A$-$A$-bimodule. Then the Hochschild homology of $A$ with coefficients in $M$, denoted $HH_\ast(A)$, is the homology of the chain complex ...
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67 views

Give an infinite sequence of principal ideals of $R$ such that the ascending chain condition does not hold

Let $R=\{\sum_{i=0}^n a_ix^i\mid n\geq 0, a_0\in\mathbb{Z}, a_i\in\mathbb{Q} \text{ for } i\geq 1\}$. Give an infinite sequence of principal ideals of $R$ such that the ascending chain condition ...
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Two ordered rings are isomorphic iff their positive semirings are isomorphic

I am looking for (a reference to) a proof that Two ordered rings are isomorphic iff their positive ordered semirings are isomorphic. The positive semiring of an ordered ring $R$ is here the ...
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31 views

Finding irreducible polynomial

1, 3 False: $f_{2^2}(x)=f_4(x)=x^3+x^2+x+1=(x^2+1)(x+1)$ is not irreducible. 2 True: Cyclotomic polynomial of order prime. I am not sure about 4. Here's my guess about 4: $x^{p^{e-1}}$ is ...
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Binary Representation of Complex Numbers

Modular arithmetic (MA) has the same axioms as first order Peano arithmetic (PA) except $\forall x(Sx \neq 0)$ is replaced with $\exists x(Sx=0)$. MA has finite models based on modular arithmetic. MA ...
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37 views

Multiplicative structure on additive group

Let $R$ be a ring without assumption of existence of unity. Let $R^{\ast} = R \oplus \mathbb{Z}$ as abelian groups. Show how to define multiplication on $R^\ast$ so that it becomes a ring with an ...
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Irreducible polynomial in $\mathbb{C} [x,y ]$

I want to prove that $t^4+xt^3+yt^2+xt+1$ is irreducible over $\mathbb{C}[x,y]$. I know that $\mathbb{C}[x,y][t]$ is a UFD just as $\mathbb{C}[x,y]$ because the base ring is a field. Can I just ...
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36 views

Every onesided nilideal of a right noetherian ring is nilpotent.

Suppose $R$ is a right noetherian ring. Prove that every onesided nilideal is nilpotent. I try to use this theorem: If R is a commutative Ring and I is nilideal of R and also I is finitely ...
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68 views

Show that Weyl algebra is noetherian

Let $k$ be a field. I want to show that the ring $D=k\left[x_1,x_2,\dots,x_n,\frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2},\dots,\frac{\partial}{\partial x_n}\right]$ which acts on ...
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Let $K$ a field with characteristic $p>0$. Show that $\{x \in K : x^{p^n} =x \}$ is a subfield.

Let $K$ a field with characteristic $p>0$. I've shown that for every positive $n$ the set $\{ x^{p^n} : x \in K \}$ is a subfield of $K$, I did this by showing that $F:K\to K: x \mapsto x^{p^n}$ is ...
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35 views

In $\mathbb{Z_2[x]}/(x^2 + x +1)$, what is $\overline{x} \cdot \overline{x}$?

In $\mathbb{Z_2[x]}/(x^2 + x +1)$, what is $\overline{x} \cdot \overline{x}$? $\overline{x} \cdot \overline{x} = \overline{x^2}$ If I divide $\overline{x^2}$ by $(x^2 + x +1)$ I get an answer of $1$ ...
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101 views

What are the elements of the quotient ring $\mathbb{Z_3}[x]/(x^3 + x^2)$?

$$R = \mathbb{Z_3}[x]/(x^3 + x^2).$$ As $\mathbb{Z_3}$ is a field we have that every polynomial in $\mathbb{Z_3}[x]/(x^3 + x^2)$ of degree less than ${x^3 + x^2}$ is a distinct element in R. So I ...
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63 views

How to establish this as being true or false?

Let $R$ and $R^\prime$ be rings such that $R$ has unity $1$ and $R^\prime$ has no $0$ divisors; let $\phi \colon R \to R^\prime$ be a homomorphism such that $\phi [ R ] \neq \{ 0^\prime\}$, where ...
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80 views

Homomorphic Compression

Can there be an algorithm such that: given a plaintextdata P, Q and compression function e: $$e(P + Q) = e(P) + e(Q)$$ $$e(P*Q) = e(P)*e(Q)$$ The idea is closely related to homomorphic encryption ...
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128 views

First Order Definitions of Finite

I would like some predicates in the language of first order Peano arithemetic (PA) that are true for the standard natural numbers and false for other types of numbers like negative numbers, fractions, ...