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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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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A question about the proof of Hilbert's Basis Theorem

I have a question regarding the proof of Hilbert's Basis Theorem. Say $I=(f_1,f_2,f_3,\dots)$ is an ideal in $A[x]$, where A is a Noetherian ring. Say we take the leading coefficients $a_i$ of all ...
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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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Need help with finding generator

$I=\{a+bi \in R\mid a \equiv b\pmod 2\}$ is ideal of $R=Z[i]=\{a+bi\mid a,b \in Z\}$. Can somebody help me to find the generator of $I$?
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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 $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?
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A questions about Group Rings

Let's say $R:=\mathbb{Z}_p[C_{p^\infty}]$ be the group ring of a Prufer group over the field of integer module a prime $p$. We have $C_{p^\infty}=\langle u_1, u_2, ..., u_n, ... |\,\,\,\, ...
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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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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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General Primality Conditions in the UFD $\mathbf{Q}(\sqrt{-d})$

Suppose $\mathcal{O}_{\mathbf{Q}\left(\sqrt{-d}\right)}$ is a UFD, so $d=1,2,3,7,11,19,43,67,163$. Are there general criteria determining whether an element in the integers of $\mathbf{Q}(\sqrt{-d})$ ...
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Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
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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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Units and zero divisors in $\mathbb{Z_2[x]}/(x^2 + 1)$ and in general

The elements of $\mathbb{Z_2[x]}/(x^2 + 1)$ are polynomials in $\mathbb{Z_2[x]}$with degree $0$ or $1$. I.e. $\{\overline{0}, \overline{1}, \overline{x}, \overline{x+1}\}$ $\overline{0} = ...
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If $R\to S$ is a ring homorphism with $J$ an ideal of $S$. Show that the preimage of $J$ is an ideal of $R$.

Let $\alpha\colon R\to S$ be a ring homomorphism. Let $J$ be an ideal of $S$, and define the preimage of $J$ by $\alpha^{−1}(J)=\{r\in R\mid \alpha(r)\in J\}$. Show that $\alpha^{-1}(J)$ is an ideal ...
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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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The integral closure of Bezout domains in arbitrary field extensions is Prüfer?

Let $R$ be a Bezout domain with quotient field $K$, $L$ an arbitrary extension field of $K$, and $\overline{R}$ the integral closure of $R$ in $L$. Is $\overline{R}$ a Prüfer domain? If the answer is ...
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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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Condition so that coefficients nilpotent implies power series nilpotent for non-noetherian ring $R$ with char$(R) = 0$

For a noetherian ring $R$ the following holds: $$\sum\limits_{i=0}^{\infty}a_iX^i \mbox{ nilpotent } \iff a_i \mbox{ nilpotent } \forall i, \mbox{ where } a_i \in R.$$ If $R$ is non-noetherian ...
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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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Weakening the assumption that an ideal is maximal

This question is from ChI, $\S{3}$ of Serge Lang's Algebraic Number Theory. Let $A$ be a commutative integral domain, integrally closed in its quotient field $K$, and let $E$ be a finite extension of ...
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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 ...