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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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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Does there exist a finite axiomatization of the quasi-algebraic theory of real matrix rings?

Some definitions. Let us take the signature of ring theory to consist of the function symbols $\{+,-,0,\cdot,1\}$ equipped with their usual airities, where the minus symbol represents a unary ...
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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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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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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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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 ...