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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twisting rings for FFT

I am implementing an FFT described by Daniel Bernstein in http://cr.yp.to/papers.html#multapps on page 332 (8 in pdf) he states the following: One can multiply in $R[x]/(x^{2n} +1)$ with $(34/3)n ...
2
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2answers
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Product and intersection of ideals in a polynomial ring

I want to show that in the polynomial ring $K[X,Y,Z,W]$ (where $K$ is a field) the equality $(X,Y)\cap(Z,W)=(XZ,XW,YZ,YW)$ holds. Obviously RHS is contained in LHS. How to show that LHS is contained ...
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1answer
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rings of polynomials over $Z_p$

An element of R is a polynomial in $x$ of degree $< r$ with coefficients from $Z_p$ (where $p$ is a prime). We use the notation $a(x)$ to represent elements of $R$. Define map $\phi :R \mapsto R$ ...
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Determining the kernel of a module homomorphism

Let $p$ be a prime and let $n$ be a positive integer such that $p^n > 2$. Set $R:= \mathbb{Z}_{p^n}$, that is, the residue ring with binary operations of addition and multiplication modulo $p^n$. ...
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1answer
15 views

Is a subring contained in the centralizer of its centralizer?

Fix a ring $B$. Given a subring $A \subset B$, we define$$A^! := \{b \in B : ab = ba,\text{ }\forall\,a \in A\},$$the centralizer of $A$ in $B$. This is a subring of $A$, so we can iterate $A^{!!} := ...
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0answers
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Non-existence of commutative rings with many nilpotent elements with some restrictions where matrix powers are efficient

Cross-posted from MO. At the moment can't find better reference than "Cycle Enumeration using Nilpotent Adjacency Matrices with Algorithm Runtime Comparisons" though certainly there are others. ...
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4answers
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Why is it necessary for a ring to have multiplicative identity?

I have read earlier that in a ring $(R,+,.)$ the following needs to hold: $(R,+)$ is an abelian group multiplication is associative and closed left and right distribution laws hold. However, I ...
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A finite von Neumann regular ring is unital and has $ab = 1$ if $ba = 1$

Let $R$ be a finite ring satisfying for any $x \in R$ there exists $y \in R$ with $xyx = x$. Show that $R$ is unital and that if $ab = 1$, then $ba = 1$. Thoughts so far: If I can show that the ...
2
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2answers
45 views

Existence of integer solution of $a^2 -17b^2 = $ any constant

When checking whether if $9-\sqrt{17}$ in the ring $\{a+b\sqrt17: a,b \in \mathbb{Z}\}$ is a prime. Suppose $$\alpha\cdot \beta = 9-\sqrt{17},$$ using norm argument $$N(\alpha)N(\beta) = ...
2
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1answer
27 views

How to create a ring in MAGMA with relations?

I'm using MAGMA221 and would like to create a ring over $GF(2)$ with respect to a list of relations. Here's what I have so far: $\mathtt{Z:=GF(2);} \\\mathtt{P<x,y,z>:=PolynomialRing(Z,3);}$ ...
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0answers
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Factor rings $R/R$ and $R/0$

Let $R$ be a ring. I want to describe the factor rings $R/R$ and $R/0$. So $R/R = \{[r]| r+R, \forall r\in R \}$ and since $r+R=R$, we get $R/ R =\{[0]\}$. And for $R/0 = \{[r]| r+0,\forall r\in ...
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0answers
24 views

There exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$.

$I\trianglelefteq \Bbb F[x]$. I want to prove that there exists an $f\in \Bbb F[x]$ such that $I=\{fg|g\in \Bbb F[x]\}$. I guess this means that I am meant to show that we have closure from the ring ...
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1answer
131 views

Does $f(x) \in \mathbb{Z}[x]$ irreducible, imply $f(2x)$ also irreducible?

It is well known that $f(x+a) \in \mathbb{Z}[x]$ is irreducible when $f(x)$ is irreducible. I was wondering whether $f(2x)$ and in general whether $f(nx)$ is irreducible as well. Though it is ...
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2answers
40 views

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$?

How does uniqueness of the additive inverse imply that $-(ax) = (-a)x$? In my title, I should be clear that the additive inverse should be unique. But how does this help? I dont even get why ...
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1answer
32 views

Minimal Polynomial Properties

If $f$ is the minimal polynomial of $\alpha$, then can there be another polynomial $g$ of smaller degree than $f$, but not monic having $\alpha$ as a root ? for example the function ...
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0answers
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Find a polynomial equation satisfied by $\phi$

I'm solving the following exercise from my class notes: Let $A=k[x^2,y^2]$ and let $M=k[x^2,xy,y^2]$ ($k$ a field). Show that $M$ is an $A$-module. Define $ \phi :M \to M$ by $\phi(m)=xym$. Find ...
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4answers
45 views

Prove that $R$ is an integral domain $\Leftrightarrow$ $R[x]$ is an integral domain

Here is an exercise(p.129, ex.1.15) from Algebra: Chapter 0 by P.Aluffi. Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain. The implication part makes no problems, ...
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2answers
123 views

Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.

Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
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0answers
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What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like?

What would the ring $\mathbb{Z}[x,y]/(x^2-y)$ look like? And also what would the ring $\mathbb{Z}[x,y]/(x^3-x-y^2)$ look like? These are two sorts of rings I have been curious about.
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1answer
41 views

Is it true that $I\cap J\subset IJ\subset I+J\subset I\cup J$

Is it true that $I\cap J\subset IJ\subset I+J\subset I\cup J$ If $R$ is a commutative ring and $I,J$ are any ideals of $R$, I don't know how the product is usually defined but I think for $IJ$ is ...
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1answer
44 views

Finding ring isomorphisms

Let $A$ be a ring with $0\neq 1$ such that $x^4=1, \forall x\in A$, with $x\neq 0$. My question is: to which ring is $A$ isomorphic? $A$ can be, for example, isomorphic to $\mathbb{Z}_2$. The ...
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0answers
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Monoid filtration

I lately been introduced to monoid filtrations and I have a couple of questions. Let $(\mathfrak{M},\star,1_\mathfrak{M})$ be a monoid with total order, $(A,+)$ the additive subgroup and ...
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2answers
41 views

Intuition for a ring homomorphism?

A map $f: A \to B$ between two rings $A$ and $B$, is called a ring homomorphism if $f(1_A) = 1_B$, and one has $f(a_1 + a_2) = f(a_1) + f(a_2)$ and $f(a_1 \cdot a_2) = f(a_1) \cdot f(a_2)$, for any ...
3
votes
2answers
64 views

Show $k[x]\cap(y-x^2,z-x^3)={0}$ in $k[x,y,z]$. $ k$ field.

Let k be a field. How could I show that $k[x]\cap(y-x^2,z-x^3)={0}$ in $k[x,y,z]$. I understand that there's a whole algorithm I could go through with Grobner basis, elimination theorem etc. but ...
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1answer
59 views

Finite Modules Isomorphism

Two vector spaces are isomorphic if and only if they have the same dimension. In particular, two vector spaces over a finite field are isomorphic if and only if they have the same cardinality. For a ...
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1answer
30 views

$R$ is a ring with identity. Why from $f(1)=0$ it's concluded that $\forall r\in R; f(r)=0$?

The original question is this: Let $R$ be a ring with identity and $\mathbb{C}$ the ring of complex numbers. Suppose $f,g:R\rightarrow \mathbb{C}$ are two ring homomorphisms such that for every $r$ ...
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1answer
183 views

The ring $\mathbb{C}[x,y]/\langle xy \rangle$

What can be said about the ring $\mathbb{C}[x,y]/\langle xy \rangle$? I was very certain that $$\mathbb{C}[x,y]/\langle xy \rangle \cong\mathbb C[x] \oplus\mathbb C[y]$$ since the elements in ...
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0answers
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Finite ring having $2^n-1$ invertible elements [duplicate]

Let $A$ be a ring with $0\neq1$ having $2^n-1$ invertible elements and at most $2^n-1$ non-invertible elements. Then $A$ must be a field? How many elements does the ring need to have? Because $2^n-1$ ...
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1answer
46 views

Linear endomorphisms of $k(t)$

Let $k$ be a field and let $k(t)$ denote the field of rational variables in $t$. Is it possible to characterize all $k$-linear transformations from $k(t)$ to $k(t)$? Is $End_{k}(k(t)) \cong k(t)$ ?
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1answer
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Prove $R$ is a finite ring [closed]

Suppose that $R$ is a ring and $D(R)$ is the set of all elements $x\in R$ such that there exist $b,c\neq 0$ with $xb=cx=0$. Prove that if $1<|D(R)|<\infty$ then $|R|<\infty$. ($|S|<\infty$ ...
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2answers
31 views

$M \neq 0$ but $M^* = 0$.

Let $A$ be a ring. For any left, resp. right $A$-module $M$ give the abelian group $\text{Hom}_A(M, A)$ the structure of a ring $A$-module (to be denoted $M^*$), resp. left $A$-module (to be denoted ...
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2answers
39 views

Formal power series over a regular ring is regular

I'm trying to prove that if $A$ is a regular ring then so is $A[[X]]$. The only proof I found of this statement is in Commutative Ring Theory by Matsumura, but it seems a bit over my knowledge so I'd ...
2
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1answer
47 views

Are all rings $\mathbb{Z}$-modules?

In my course of associative algebra we covered modules and an excercise involved showing that every ring $R$ can also be viewed as an $R$-module. This was straightforward enough. Is it also true that ...
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0answers
28 views

Commutative rings of order $p^3$. [duplicate]

Does anyone know of a listing of all rings (or at least those that are commutative) of a given order? In particular $p^3$, for a prime $p$.
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If $A$ is a noncommutative ring then all biderivation is inner.

Before the question I will post some definitions: Derivation: An additive map $\delta: A \longrightarrow M$, where $A$ is a ring and $M$ is a $(A,A)-$bimodule is called derivation if $\delta(xy) = ...
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0answers
55 views
+50

A Problem for Nil-Ideals

Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for non-commutative rings I think this may not be true. ...
2
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0answers
48 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
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1answer
39 views

gcd of $x$ and $2$ in $Z[x]$

In $Z[x]$, $x$ and $2$ has gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2.f(x)+x.g(x)$ we are supposed to arrive at a ...
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1answer
28 views

Every ring with $1$ and with no zero divisors and no non-trivial ideals is a division ring

It is well known that every commutative ring with unity $R$ that contains no non-trivial ideal is a field, since given $a \neq 0$, $(a)=R$, therefore there exists $x \in R$ with $ax=xa=1$. What ...
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1answer
30 views

Non-artinian center

Recall that an Artinian ring is a ring that satisfies the descending chain condition on ideals. What is an example of an Artinian ring whose center is non-Artinian?
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1answer
31 views

Jacobson radical of tensor product

Suppose $R$ and $S$ are associative rings with unit and that $J(R)$, the Jacobson radical of $R$, is a nil ideal. Clearly if $R$ is commutative then $J(R)\otimes_\mathbb{Z} S$ is a nil ideal. Is this ...
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0answers
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Composition of module homomorphisms and their kernels

Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module ...
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0answers
28 views

Kaplansky characterization of principal Artin ring

I would like to learn the proof of this paper of Kaplansky where it is proven that for a commutative ring every module split as sum of cyclic module iff the ring is an Artin principal ideal ring (well ...
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0answers
33 views

Unit group of an imaginary quadratic ring

Let $R$ be an imaginary quadratic ring. Then, the unit group $R^{\times}$ is finite. To prove this, I worked with normal forms, algebraic integers and the fact that $R \not \subset \mathbb{R}$. But I ...
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3answers
85 views

For a prime integer $p \in \{2, 3, 5, \cdots\}$, is $pR$ a maximal ideal in $R$?

If $R$ is a commutative ring with unit and $p$ is a prime number ($2,3,5,\cdots$), then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?
3
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1answer
53 views

Prove that $f_n=-1+\prod_{i=1}^{n}(X-i)$ is irreducible in $\mathbb{Z}[X]$

Prove that, for all $n\in \mathbb{N}$, $f_n=-1+\prod_{i=1}^{n}(X-i)$ is irreducible in $\mathbb{Z}[X]$.
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2answers
72 views

Proving the Ideal Generated by the Coefficients of $f(X)\cdot g(X)\in R[X]$ is $R$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the ...
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1answer
59 views

If $[A,A]A[\lambda,A] = 0$ then $\lambda \in Z(A).$

Suppose that $A$ is a unital ring and $([A,A]) = A.$ If $[A,A]A[\lambda,A] = 0$ prove that $\lambda \in Z(A).$ Comments: This is part of an exercise I'm doing, I'm posting this part because I am ...
3
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0answers
24 views

Module endomorphisms with the same kernel

Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in ...
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0answers
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Reference for a Dickson Determinant Polynomial

For $2\leq \ell \leq k$, consider the polynomial \begin{equation} P_{k,\ell} = \prod_{1\leq a_1+\ldots+a_k\leq \ell} (a_1x_1+\ldots + a_kx_k)\in \mathbb{F}_2[x_1,\ldots, x_k] \end{equation} ...