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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Determining prime ideals lying above a given ideal

Let $R=\mathbb{Z}[x]/(f)$, where $$f(x)=x^4+42x^3-11x^2+22x-2002002002002002.$$ Let $I=3R$, the ideal generated by $3$ in $R$. Find all prime ideals of $R$ that contain $I$. I am hoping to get ...
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Is ${(k^n)}^{\otimes r}$ a faithful $k\Sigma_r$-module for $n\geq r$?

I have the following question: Let $k$ be an infinite field. Let $E:={(k^n)}^{\otimes r}$ and let the symmetric group $\Sigma_r$ act from the right on $E$ by place permutations. It is well-known that ...
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Short proof of the coincidence of left dom.dim., right dom.dim. and relative dom.dim. for semi-primary left QF-3 rings?

is there a short and/or elementary proof of the following fact (which is taken from theorem 7.7 from H. Tachikawa, ‘‘Quasi-Frobenius Rings and Generalizations’’, Springer Lecture Notes in Mathematics):...
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Dominant dimension $\geq 2$ implies a certain double centralizer property

let $A$ be an Artin algebra and $M$ in $\mathfrak{mod}\ A$. Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent: $\bullet$ $Ae$-dom.dim.$(A)\geq 2$ $\...
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Multiplicative Inverse of Polynomials in Finite field

Find the multiplicative inverse of $x + 2$ in the field $\Bbb Z_5[x]/(x^2 + 2)$. I have done the following so far: \begin{align*} x^2+2 &= (x+2)(x+3) + 1\\ (x+2)(x+3) &\equiv -1 \pmod {x^2+2}...
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Left ideals in a subring of $M_2(R)$

Let $R$ be a 2-dimensional complete regular local ring $R$ over an algebraically closed field $k$, that is $R\cong k[[x,y]]$. Now look at the the following subring $A$ of $M_2(R)$: $A=\begin{pmatrix} ...
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Local ring at generic point

Let $X$ be a smooth projective variety, and $Y$ a subvariety of codimension one (both are irreducible). I want to show that the local ring $\mathcal{O}_{Y,X}$ at the subvariety $Y$ (which is nothing ...
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Degree of the minimal polynomial of the sum of two integral elements over a UFD

Let $D$ be an integral domain ($D$ is a noetherian UFD, if necessary) and let $a,b$ integral over $D$. Let $f$ be the minimal polynomial of $a$ over $D$ and assume it is of degree $n>1$, and let $g$...
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Irreducible is prime in the ring $\mathbb{Z} + X\, \mathbb{Q}[X]$

If a ring is an UFD, its irreducible elements are exactly its prime elements. Show that the reverse is not true. Give a nontrivial counterexample. Hint: Consider the ring $\mathbb{Z} + X\, \...
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Tensor product of algebras and generating sets.

Let $A$ be a module over $k$ generated by $x$ and $y$. The generating set for $A \otimes_k A$ is $\{x \otimes x, x \otimes y, y \otimes x, y \otimes y\}$. But does this still hold if $A$ is an algebra ...
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Idempotents and symmetries of Zn

[Soft question out of curiosity] While reading about idempotents of a ring, I used $\mathbb Z_n$ as a convenient example. In visualizing the ring's structure, I was intrigued by strange symmetries ...
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Generalisation of a Theorem of McCoy to rings with bounded index of nilpotency

Let $R$ be a commutative ring. I have seen that $c_0+c_1x+c_2x^2+\cdots.+ c_nx^n \in R[x]$ is nilpotent iff $c_i$ is nilpotent for all $i = 0,1, ... ,n$. But similar result is not true for $R[[x]]$. ...
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If $a^{2k}=a$ for all elements of a ring then $a= -a$ for all $a$

If there is a positive even integer $n$ such that $a^n= a$ for all elements of a ring R then how to prove that $a+a=0$ for all $a$ in R. For Boolean ring, it is easy. But how to prove for $n>2$.
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Find the $\ker(f)$ and $\text{Im}(f)$.

Consider the rings $\mathbb{Z}$, $\mathbb{Z}_{4} = \{\bar{0},\bar{1},\bar{2},\bar{3}\}$ and $\mathbb{Z}_{12} = \{[0],[1],[2],...[11]\}$. Define $f: \mathbb{Z} \to \mathbb{Z}_{12}$ by $f(x) = 9x$. ...
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Efficiently computing GCDs in $\mathbb{Z}[(1+\sqrt{-19})/2]$

The ring $\mathbb{Z}[(1+\sqrt{-19})/2]$ is a PID; hence any two elements have a GCD. How you would compute their GCD? In a Euclidean domain, you would use the Euclidean algorithm. But $\mathbb{Z}[(1+\...
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Is $\mathbb{Z}[x]/(x^3-2)$ a field?

Exercise: Prove or disprove: $\mathbb{Z}[x]/(x^3 − 2)$ is a field. I would say, that it is not a field. We can use the following theorem: Let $R$ be a commutative ring with identity and $M$ an ...
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Is the power of a finitely generated associative algebra still finitely generated?

Let $A$ be an associative finitely-generated algebra. I want to show that any power of $A$ is finitely-generated as well. Definition of power: $$A^k \stackrel{\text{def}}= \text{ { all finite ...
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Rings with bounded index of nilpotency are Dedekind-Finite

Recently in an article by A. A. Klein I have seen this result: A ring $R$ with bounded index of nilpotence is Dedekind-Finite. Can anyone help me prove this result?
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Counting rings of order $p^3$

MathWorld states that there are exactly 52 rings of order 8 (multiplication in rings may be not commutative and perhaps there will be no neutral element) and 53 rings of order $p^3$ where $p$ is an ...
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finite field extensions: how to compute norm and trace

I'm studying abstract algebra and I'm stuck in the topic of fields. I don't understand what the following definition Let $R$ be a commutative ring and let $S$ be a commutative $R$-algebra, which is ...
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Can $\operatorname{Spec}(A)$ be expressed as an inverse limit?

We know that given a ring $A$ such that $A/\mathfrak{R}$ is absolutely flat, then $\operatorname{Spec}(A)$ is Hausdorff (it's an equivalence). So $Spec(A)$ becomes a quasi-compact, Hausdorff and ...
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To prove ; $pa:=a+a+… p $ times $=0 , \forall a \in R$ and $a^p=a , \forall a \in R$ for some prime $p$ then the ring $R$ is commutative

If in a ring $R$ , $\exists $ prime $p$ such that $pa:=a+a+... p $ times $=0 , \forall a \in R$ and $a^p=a , \forall a \in R$ , then how to prove that $R$ is commutative ? I would not want to use ...
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Example of finite ring which is not a Bézout ring

A left (or right) Bézout ring is a ring in which any two elements generate a principal left (resp. right) ideal. Assume that we have a finite ring R. Does there exist some classification theorem (...
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Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
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Henselization of the ring of polynomials

I am trying to understand example of Henselization from wiki. http://en.wikipedia.org/wiki/Henselian_ring#Henselization It says that Henselization of the ring of polynomials localized at point $(0, \...
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prove that $(E_{p^n},*)$ is cyclic group

if $p \in$ $\mathbb{N}$ is a prime integer, how can i prove that $E_{p^n}$ the group of invertible elements of $\frac{\mathbb{Z}}{p^n\mathbb{Z}}$ is a cyclic group.
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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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Non-UFD that there exists set $X$ of any cardinality and $a$ that $xy$ is not divisible by $a^2$ for any $x,y \in X$ and $x^2$ is divisible by $a^2$

Let's say we want to construct a non-UFD that is a commutative ring that satisfies: There exists $a$ such that there exists a set $X$ of elements of finite cardinality $k$ such that for any $x,y \in ...
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Find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$.

Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two. If we consider a rational function field $k(x,y)...
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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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Is the completion of a Dedekind domain a PID?

Ths is a basic question on Dedekind domains. Let $R$ be a Dedekind domain, $P$ a non-zero prime ideal of $R$. I know that the localization $R_P$ is a PID, but is it true that the completion $\hat{R}...
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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, $N(...
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Finding the prime element of a place of a function field of a elliptic curve.

Let $p$ be an elliptic curve in $\mathbb{C}[X,Y] $. Consider the quotient ring $A = \mathbb{C}[X,Y]/(p) $ and its field of fractions $F = frac(A) $. For all $f + (p) \in A$, define $deg_A(f + (p))= ...
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Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b \in \...
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Any simple proof of that localization of a UFD is a UFD without using Kaplansky condition?

Using Kaplansky condition, the proof of this statement is quite easy. Before knowing this condition, I tried to prove this statement by actually showing the definition of UFD holds on $S^{-1}A$ ($A$ ...
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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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$p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$

I need to show that $p(x)=x^4-2x^2-4$ is irreductible over $\mathbb Q.$ Here's what I've done: Please tell me if it's correct Over $\mathbb C,$ $x^4-2x^2-4\\=(x^2-1)^2-5\\=(x^2-1+\sqrt 5)(x^2-1-\...
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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, ... |\,\,\,\, u_1^p=1,\,\...
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A Lemma of Kaplansky

Source: Rings With a Polynomial Identity, Irving Kaplansky The Lemma: Suppose that $\mathbb{A}$ is an $\mathbb{F}$-algebra, where $\mathbb{F}$ is a field. Then, suppose that $\mathbb{A}$ ...
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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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A Commutator Identity in Rings

In a ring (or associative algebra), let the commutator $[A,B]$ be defined as $[A,B]=AB-BA$. I have asked earlier for a general formula for the expression $[x_1\cdots x_m,y_1\cdots y_m]$ in a group $...
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Counterexamples to the Artin-Rees Lemma

This well known Lemma about $I$-stable filtrations asserts: Lemma (Artin-Rees) Let $A$ be a Noetherian ring and $E$ a finitely generated $A$-module. Let $F$ be a submodule of $E$ and $\{E_i\}$ an $I$-...
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$G \simeq R^{\times}$

What is known about the groups G for wich there exist a unitary ring R, such that $R^{\times} \simeq G$? I can easily prove that The only G cyclic with this property(Edit:and odd order) are those who ...
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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 $f_{1},......
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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 $i\in\lbrace 1,\dots,n\rbrace$....
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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} = \overline{...
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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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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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About quotient ring

I want to find the value $|\mathbb{Z}({\sqrt{2})/(3+\sqrt{2})}|,|\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}|$ and also the number of ideals of $\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}$. But still not ...
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When every matrix is a sum of nilpotent (idempotent or invertible) matrices ??

Let $R$ be a ring with non-zero identity. Consider the following three properties: Every $A \in M_n(R)$ is a sum of nilpotent matrices. Every $A \in M_n(R)$ is a sum of idempotent matrices. Every ...