Questions tagged [finite-rings]

Use with the (ring-theory) tag. The tag "finite-rings" refers to questions asked in the field of ring theory which, in particular, focus on rings of finite order.

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Let $R=\mathbb Z_6\times \mathbb Z_8$. Find all those $x,y\in R$ such that $Rx+Ry=R$.

I am stuck on the following question: Let $R=\mathbb Z_6\times \mathbb Z_8$. Find all those $x,y\in R$ such that $Rx+Ry=R$. Now if $x$ or $y$ is a unit in $R$ then $Rx+Ry=R$ will hold trivially. How ...
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Solveablity of Diophantine equation over "computer numbers"

Hilbert's tenth problem asks whether there is an algorithm to determine if a given solution set to a Diophantine equation is non-empty. There is no such algorithm. In practice for many engineering ...
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Structure Theorem for non-abelian finite groups or rings

How many structure theorems do we have in Abstract Algebra for finite algebraic structures? I know some of the following theorems: If $G$ is a finite abelian group, then $G$ is a product cyclic ...
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On the definition of cohomological dimension

Let $G$ be a group and $R$ a commutative unital ring. We define the $R$-cohomological dimension of $G$ to be $$cd_R(G) := \sup \{ n : H^n(G, M) \neq 0 \text{ for some } R[G]\text{-module } M \}.$$ I ...
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Let $(R, +, \cdot)$ be a finite ring without zero divisors, show that $R$ has a neutral element for $\cdot$. [duplicate]

I have to prove the question in the title, but I am having some difficulties. Here's a sketch what I've already tried: Choose $a \in R$. Because $R$ is finite, there exist positive integers $i$ and $j$...
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Finite quotients of ring of integers of local field

Let $K$ be a non-Archimedean local field, so either a finite extension of $\mathbb{Q}_p$ or a finite extension of $\mathbb{F}_q((t))$. Let $\mathcal{O}$ denote its ring of integers and $\pi$ a ...
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258 views

If $x \in R$ is non-invertible implies $x^2 \in \{\pm x\}$ and $|R| >9$ odd then $R$ is a field

Let $(R, +, \cdot)$ be a commutative ring with $2n+1$ elements, for some $n\neq 4$ a positive integer. Suppose also that $R$ also satisfies the following condition: If an element $x\in R$ is non-...
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Polynomials for which the induced polynomial map is zero

Let $R$ be a commutative ring with $1$. Out of curiosity, I wonder what is the state of art about $I_R=\{P\in R[X]\mid P(r)=0 \mbox{ for all }r\in R\}$. This an ideal of $R[X]$, which can be rewritten ...
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Classify all finite rings such that each unit has order 24

Problem: Suppose $R$ is a finite (associative) ring with 1 such that every unit of $R$ has order dividing 24. Classify all such $R$. My attempt: I had to quotient out the jacobson radical $J(R)$ so ...
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solutions to $u^2 + u = 0$ in quotient ring $GF(2)[x]/p(x)$

In a finite field $GF(2^n)$, $u^2+u=0$ has only two solutions: $u=0, u=1$. (Not sure I can prove why this is true, probably something to do with invertibility.) In a quotient ring $GF(2)[x]/p(x)$ ...
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Question related to Ideals of the ring of all functions from the set $\{1,2, \dots , 10\}$ to $\mathbb Z_2$. [duplicate]

I have the following question in a competitive exam , but I failed to answer it.The question is: Let $\mathcal R = \{f:\{1,2, \dots , 10\} \rightarrow \mathbb Z_2\}$ be the set of all $\mathbb Z_2$...
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1 vote
Let $R$ be a finite commutative ring with unity ; then does there exist a non-empty proper subset $A \subseteq R$ and $f(X) \in R[X]$ such that $f(r)=1 , \forall r \in A$ and \$f(r)=0 , \forall r \...