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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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 $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?
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Help needed in understanding a proof

Claim: Let $M$ be a $R$-module ($R$ is an integral domain) and $p \in R$ be a prime. Suppose there exists non-empty finite subsets $B$ and $C$ of $M \backslash\{0\}$ such that $M= \bigoplus_{m \in ...
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Associated primes and their heights

Let $(S,m)$ be a commutative Gorenstein local ring, $I$ an ideal of $S$ such that $\operatorname{ht} I=t$, and $R=S/I$. Let $a \in m$ be an $R$-regular element such that for any prime ideal ...
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some ring theory applied to holomorphic functions

I'd like to know if my understanding of this business is correct. Let $U \subset \mathbb{C}^n$ be open and connected. The set $\mathcal{K}(U)$ of meromor­phic functions on $U$ is a field. Is it true ...
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Fields and quotient ring

Let $P(X)\in{\mathbb{R}[X]}$ irreducible polynomial. Then $\mathbb{R}[X]/(P(X)=X^2+1)\approx{\mathbb{C}}$. If $P(X)=X^2+X+1$ also $\mathbb{R}[X]/(P(X))\approx{\mathbb{C}}$? Or for a arbitrary ...
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Properties of cyclotomic polynomial

Assume first that $p$ a prime divides $n$. I have to show that $\Phi_{np}(X)=\Phi_n(X^p)$. Here is what I tried: Suppose $\eta_i$ are roots of $\Phi_{np}(X)$ so $\eta_i=\text{exp}(\frac{2\pi i ...
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Give an example showing that $\phi(R^\times)$ may be strictly smaller than $S^\times$.

Let $R, S$ be commutative ring and $\phi$ a surjective ring homomorphism from $R$ to $S$. Give an example showing that $\phi(R^\times)$ may be strictly smaller than $S^\times$. Any help is ...
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If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$

If there is an integer $n$ such that $n^2\equiv3\pmod p$, where $p$ is prime, prove there are integers $a$ and $b$ such that $|a^2-3b^2|=p$. So $n^2-3 = pm$ for some integer $m$, and I know $|a^2 ...
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What is the fraction field of formal power series ring over a field $F$?

The field of fractions of the formal power series ring $F[[x]]$ over a field $F$ can be obtained by inverting the elements $x$. Let $R=F[[x]]$. I have difficulty in finding the isomorphism $$ ...
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207 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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quadratic rings of integers vs cubic rings of integers in number fields

I would appreciate if someone could give me some clues about cubic $\mathbb{Z}$-rings of number fields. So far I have only learned about quadratic rings and I would like to see if there are any ...
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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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Multiplicative group into ring operation

My question is simple, though it proves to be much more difficult than it sounds. Suppose I want to find a binary operation to add extra structure to a multiplicative group (so it becomes a ring). ...
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Different elements in a factor ring

Studying for my algebra exam I found the following problem, which I'm not sure how to solve Let $f = X^2 + 1 \in \mathbb{F}_5[X]$, $R = \mathbb{F}_5[X]/\langle f \rangle$ and $\alpha = X + \langle ...
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Isomorphism, integers mod n and the chinese remainder theorem

This is an extension of my previous question: isomorphism, integers of mod $n$. Setup: If $n = p_{1}\cdot p_{2} \cdots p_{n}$ where $p_{i}$ distinct primes for all $i\in\lbrace ...
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What's the connection between irreducible polynomials and fixed-frobenius elements in a finite ring?

Consider the ring $A_{p,n} = \mathbb{F}_p [x]/ (x^{p^n}-x)$. It has a basis $\{1, x, x^2, \ldots, x^{p^n - 1}\}$. The Frobenius endomorphism $x \mapsto x^p$ permutes elements of this basis. I've ...
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Using formal power series.

Suppose $R$ is a commutative ring (with or without $1$), and for $a_0,...,a_m,b_0,...,b_m\in R$ and for all $0\le k\le m$ we have $$a_0b_k+a_1b_{k-1}+\dots+a_kb_0=0 $$ then there's some nonzero $r\in ...
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“Rule's” for reducilibility depending on degree of a polynomial

I want to make sure I have the following information correct. Here is what I understand regarding the reducibility of polynomials of different degrees on $F[x]$, $F$ a field. Let $f(x) \in F[x]$ f ...
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Ring Integration

In thinking about various methods of integration, I began to wonder if there was some sort of unifying theory relating integration and ring theory. For example, would there be a way to make sense of ...
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Rings and Categories of Modules by Anderson and Fuller: Corollary $7.4$

I am reading the book Rings and Categories of Modules by Anderson and Fuller. I don't understand corollary $7.4$ of that book. Can anyone explain that corollary to me? Thank for any insight.
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homework: rings, matrices and polynomials

$A,B$ are both $n \times n$ and diagonal matrices. Prove that there is a matrix $X$ which is $n \times n$, and polynomials $p$ and $q$ such that $A= p(X), B= q(X)$ Is this true for ANY 2 matrices (we ...
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Minimal spectrum of graded rings

Let $R$ be a left Noetherian ($\mathbb N$-)graded ring and let $R_0$ be its $0$-th component. When $R$ is commutative it is well-known, and easy to prove, that the minimal prime ideals of $R$ are ...
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Help about trace of a module

I'm reading about Trace and Reject, which definition: $Tr_M(U)= \sum \lbrace Im(h)|h\in Hom_R(U,M) \rbrace$ $Rej_M(U)= \bigcap \lbrace Ker(h)|h \in Hom_R(M,U)\rbrace$ So is that right: if ...
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Quotient of complete linearly topologized ring

The quotient of a complete metrizable group by a closed normal subgroup is always complete, but there are examples to show this need not be true for non-metrizable groups. Here complete means every ...
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determine the roots of $X^{2}-1$ in $\left(\mathbb{Z}/36\mathbb{Z}\right)[X]$

I have no clue what to do here. Any tips/hints? I already got 1 and -1=35 Because $(-a)^{2}-1=(a)^{2}-1$ i only need to check the first 18 digits. And because $(2k)^2 - 36 \cdot q \neq 1$ the even ...
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Semirings (or Rigs)

Can anyone give me some famous quotients of semirings or Rigs (i.e. semirings described by generators and relations)? For example M. Fiore and T. Leinster have given the normal form for the elements ...
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Computing ext over graded rings

This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that $A$ is a Koszul ring (for the definition of Koszul ring ...
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Product of ideals closed set

Let $R$ be a topological ring (in fact $R$ is metrizable) and let $I,J$ denote ideals of $R$. Suppose also that they are closed with respect the topology of $R$. Is it always true that the product ...
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If $\mathbb{C}[G]$ is Noetherian and $G$ has a representation on $V$, when must $V$ be finite-dimensional?

I know this is a bit vague, but please bare with me here. Let's assume that $G$ is a finitely-generated torsion group. I want to show that $G$ is a finite group if I add some conditions. I suspect ...
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Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
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A question about the consequence of Prime Avoidance.

I have found the following statement: Let $R$ be a Noetherian ring and $x$ is a non-zero divisor of $R$. Let $P$ be a prime ideal associated to $xR$. Then by Prime Avoidance there exists a non-zero ...
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A criterion for finite generation of subalgebras of a polynomial ring

In her 1926 paper on invariant theory, Emmy Noether uses a certain "finiteness criterion" which I wish to translate and maybe find a more modern reference to. The original wording is: Ein ...
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Functors that have a natural Isomorphism

Find different functors $T, S: Rng \rightarrow Rng$ both identity on objects IE: for each ring $R, T(R)=S(R)=R$, such that there is a natural isomorphism between T and S. I know that a natural ...
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In $ℤ/Nℤ$, which units are successors to zero divisors?

What are the units $x$ in $ℤ/Nℤ$ of the form $x = 1 + \overline{kd}$ for a divisor $d$ of $N$ and $k ∈ ℤ$, i.e. $$U_N[d] := \{x ∈ (ℤ/Nℤ)^×;\; ∃ k ∈ ℤ : x = 1 + \overline{kd}\} = \ker \big((ℤ/Nℤ)^× → ...
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Krull dimension of a module over the Weyl algebra

Let $A_{n}$ be the $n$th Weyl algebra over a field, with generators $x_{1}, \ldots, x_{n}, y_{1}, \ldots, y_{n}$. Why is the Krull dimension of the rigth $A_{n}$-module $A_{n}/x_{1}A_{n}$ equal to ...
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Prove that $[L,Rad(L)] \subseteq N$ for finite-dimensional Lie algebra $L$

I need to prove the following fact: if $L$ is a finite-dimensional Lie algebra over field of characteristic $0$, $Rad(L)$ is its radical, and $N$ is the maximal nilpotent ideal in $L$, then ...
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Frobenius from Hurwitz's theorem

Can we deduce Frobenius theorem from Hurwitz's theorem on Normed division algebra? Frobenius theorem states that the only associative finite dimensional division algebras over the real numbers are R, ...
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A commutative ring with alternating and commutativity properties with infinite distinct elements

Is there any nontrivial commutative ring without multiplicative identity that satisfies alternating property ($x \cdot x = 0$ for all $x$ where $\cdot$ is multiplication operator and $x \cdot y \neq ...
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verifying if an ideal is prime

Let $R=\mathbb{Z}[\sqrt{-5}]$ and let $\mathfrak{i}=(2,1+\sqrt{-5})$ the ideal generated in $R$ by $2$ and $1+\sqrt{-5}$. I want to prove that $\mathfrak{i}$ is prime. So i considered the surjective ...
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Why $\text{top}h$ is an isomorphism?

I am reading the book Elements of representation theory of associative algebras I have a question about from Line -9 to Line -6 of page 29, the proof of Theorem 5.8. How to show that ...
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Left ideals of central simple algebra generated by symmetric element

Let $F$ be a field of characteristic $\neq 2$. Suppose $(A,\sigma)$ is central simple algebra over a field $F$ with an orthogonal involution. Assume $(A,\sigma)$ is non-split. How to show that every ...
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A necessary and sufficent condition for a ring to be a UFD

I came across the following question in Hungerford's Algebra: An integral domain $R$ is a UFD iff every non-zero prime ideal contains a nonzero prime principal ideal. The forward direction is ...
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Non-commutative integral extensions?

In Commutative algebra there is a notion of an integral extension: Let $P$ be a subring of $R$. Then $R$ is the integral extension of $P$ if each element of $R$ is a root of a monic polynomial with ...
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Multiplication structure for finite abelian rings of order $p^2$.

Consider an odd prime $p$ and a positive integer $a$ as close to 2 as possible that is not a quadratic residue $mod$ $p$. If we extend the ring $mod$ $p$ with the element $b = a^{\frac{1}{2}}$ then ...
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Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
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Projective dimension of simple module

Let $R$ be a commutative ring and $M$ a simple $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. Then it is known that $$ \mathrm{pdim}_{R}(M)=\mathrm{pdim}_{R_{\mathfrak{m}}}(M), $$ ...
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Writing out an isomorphism between two rings

I am having a hard time writing a bijective map between the two rings: $$ R = \dfrac{k[x,y,z,u,v]}{\left<(x-y)z+uv\right>} \cong \dfrac{k[x,y,z,u,v]}{\left<(x-y)z\right>} = S. $$ I ...
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$k[V]^G = \widetilde{A}$ where $\widetilde{A}$ is the normalization of $A$

Let $V$ be a finite dimensional vector space over $k =\overline{k}$ and let $G$ be a subgroup of $GL(V)$ so that $k[V]^G$ is finitely generated. Let $A$ be a subring of $k[V]^G$ that is finitely ...
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What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,…]$

What are the prime ideals of $\mathbb{R}[x_1,x_2,x_3,...]$? (this is the ring of polynomials over the reals with countably infinite many indeteminates). My attempt: I think taking the principal ...
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Degree 1 elements in a graded ring from a blow-up perspective

This may be an elementary question but I hope this question will benefit others as much as myself. Let $k^4 = Spec \; k[x_1, x_2, x_3, x_4]$. Writing $Bl_{\mathcal{I}}(k^4)$ as $R = ...