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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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 ...
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Why is $A/I^n A \cong \hat{A}/I^n \hat{A}$?

Let $A$ be a commutative ring, $I \subset A$ a finitely generated ideal. Define $\hat{A} := \varprojlim A/I^n$. What is the best way to proof that $A/I^nA \cong \hat{A}/I^n \hat{A}$ for all $n$?
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Complex matrices: looking for homomorphism

Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by $ f(a+bi) = ...
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Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. 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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61 views

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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Showing that $\frac{\mathbb{C}[X]}{<x-1>}$ is isomorphic to $\mathbb{C}$

I'm trying to show that $\frac{\mathbb{C}[X]}{<x-1>} \cong \mathbb{C}$ and I am not sure if this argument is correct. Define $\phi: \mathbb{C}[X] \to \mathbb{C}$ by $\sum a_it^i \to \sum a_i$. ...
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How to define average operator on the Real (semi) ring

How does one define the average operator on groups/ring/filed of Real numbers with multiplication (and/or addition) operator $(\Re, ., +)$? Is this extendible to for example the semiring of $(\Re, ...
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29 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
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Does This Ring have a Name?

Let $M_1=\{0,1,2,4,5,8,9,10,\cdots\}$ be the set of nonnegative integers that can be written as a sum of two perfect squares. Let $M_2=\{\sqrt{m}: m\in M_1\}=\{0,1,\sqrt{2},2,\sqrt{5},\cdots\}$. Let ...
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Showing isomorphism between quotient rings

Let $f: R_1 \rightarrow R_2$ be a surjective ring homomorphism. Let $I_1$ be defined: $I_1=\{r_1\in R_1| f(r_1)\in I_2\}$, and I've showed that $I_1 \lhd R_1$. It's also known that $I_2 \lhd R_2$. ...
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Evaluation maps over polynomials

just looking for feedback and/or hints about this proof I've been working on. No answers please, but I'd like to know if I'm on the right track here. So I'm given a field $F$, and a non-zero $n ...
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Is $\min \deg$ in a dirichlet subring interesting or is it always $1$?

Let $s \in C$. Let $D = A[[n^{-X}]]$ be a subring of the formal (or absolutely converging on a region; whatever is needed) Dirichlet series with base ring $A$. Define a minimal Dirichlet series for ...
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“Filters” of associative rings

Does there exist some good notion of "filters" for arbitrary associative rings with unity that would generalize filters of Boolean rings? For me, if $R$ is an associative ring with unity $1$, it ...
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Resolution of module over polynomial ring

The problem is: Let $F$ be a field, and let $R = F[x_1, \ldots, x_r]$, the polynomial ring over $F$. Consider the $R$-module $M = R/(x_1, \ldots, x_r) \cong F$. Find a resolution of $M$ by free ...
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For all ideals $I_1,I_2$, if $S^{-1}I_1 = S^{-1}I_2$ (localizations) then $I_1 = I_2$?

The wiki page says that the above implication holds if the ideals are prime. Here the multiplicative set $S$ contains $1$ but not $0$ and we are on a commutative ring $A$. What can we say about the ...
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Ideals - A Geometric Interpretation?

The standard way to define an ideal is as follows: $I$ is an ideal if it satisfies the following conditions: $(I,+)$ is a subgroup of $(R,+)$ $\forall x \in I$, $\forall r \in R :\quad ...
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Classify all possible $R$-module structures on a vector space

Let $V$ be an $n$-dimensional complex vector space. In particular, it is an Abelian group. Let $R$ be a (commutative, unitary) $\mathbb C$-algebra. Problem. I would like to parameterize all ...
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$(a)(b)\subset (ab)$ or $(ab)\subset (a)(b)$?

Let $R$ be a commutative ring. Consider an ideal $(a)$ generated by $a\in R$. Note that $(a)=\{ra+na : r\in R, n\in \textbf Z\}$ since $R$ has no identity. I wonder if $(a)(b)\subset (ab)$ or ...
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Homomorphic image of integral extension is integral extension.

I am working in one of the problem if ring extensions. Let $S$ be an integral extension of $R$ and $f:S\rightarrow{S}$ is a ring homomorphism such that $f(1_{S})=1_{S}$ then $f(S)$ is an integral ...
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Functional equation on a ring

Let $R$ be a ring with identity. Find all functions $f: R \to \Bbb R$ satisfying the functional equation \begin{equation} f(xp+yq, yp+xq)=f(x, y)f(p, q) \end{equation} for all $x, y, p, q\in R$, and ...
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37 views

Module is free of finite rank $\implies$ submodule is free of finite rank?

Let $M$ be $R$-module, where $R$ is commutative ring with $1,$ and $N$ be submodule of $M.$ If $M$ is free of finite rank, so is $N \ ?$ Answer: False. Let $R=M=\mathbb{Z}_{6}$ and ...
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Prove that a(mn)=a(m)a(n), (n,m)=1

Given a positive integer $n$ where $a(n)$ is the number of non-isomorphic abelian groups of order n. 1) Prove that $a(mn)=a(m)a(n), (n,m)=1$ 2) Prove that $a(p^k)$ is the number of partitions of k, ...
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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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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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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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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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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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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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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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104 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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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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39 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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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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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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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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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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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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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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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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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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36 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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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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88 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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105 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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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 ...