# 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.

190 questions
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### Stuck: Finding an Isomorphism for an Invertible Ring

I'm stuck on a problem creating an isomorphism between rings. Specifically, let $\mathbb{Z}[\sqrt{7}] = R$. Then for the invertible group $(R/3R)^\times$, I want to find an isomorphism to another ...
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### Do there exist finite commutative rings with identity that are not Bézout rings?

A similar question has been asked before: Example of finite ring which is not a Bézout ring, but has not been answered. There also seems to be a dearth of resources online regarding this ...
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### If $R$ is a finite ring, then $\exists_{n>m>0}: x^n=x^m$ for all $x\in R$

I need some help for the following proof: If $R$ is a finite ring, then $\exists_{n>m>0}: x^n=x^m$ for all $x\in R$. I feel there's one or more little tricks to use to see how you get to ...
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### A finite ring which contains a field [closed]

Prove or disprove: If $A$ is a finite ring such that there exists a field $K,$ $K \subset A,$ then $|A|$ is a power of $|K|.$
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### Let $R$ be a finitely generated subring of a number field. Is $R/I$ finite for every non-zero ideal of $R$?

Given any finitely generated subring $R$ of a number field (finite extension of $\mathbb{Q}$) or a global function field (finite extension of $\mathbb{F}_p(T)$), does $R$ have the property that $R/I$ ...
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### Show that $\Bbb Z_p[i]$ is isomorphic to $\Bbb Z_p[x]/\langle x^2+1\rangle$.

Let $p$ be a prime number. Show that $\Bbb Z_p[i]$ is isomorphic to $\Bbb Z_p[x]/\langle x^2+1\rangle$. My attempt: Define $$\phi : \Bbb Z_p[x] \to \Bbb Z_p[i]$$ by $\phi\big(f(x)\big)=f(i)$. ...
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### Factor $x^ 5 - x^4 - x^ 2 - 1$ modulo $16$, and over $\mathbb{Q }$

From ARTIN algebra books chapter $12$ question $4.19$: Factor $x^ 5 - x^4 - x^ 2 - 1$ modulo $16$, and over $\mathbb{Q }$ My works : I have check in $\mathbb{Z}_{16}$ as $x^ 5 - x^4 - x^ 2 - 1$ is ...
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### Enumerating finite local commutative rings effectively

Background I am interested in writing a program to extend the sequence Number of commutative rings with 1 containing n elements. This sequence is not as well linked-to as the other three of its type, ...
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### Examples of Finite Non-Unital Integral Rings [duplicate]

Here, integral rings are rings without nonzero zero-divisors. In this question, rings are not assumed to be unital (i.e., they may not have the multiplicative identity). Some people call them rngs. ...
1answer
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### How to draw a sublattice to exhibit diagonalization?

Given the matrix: $$A=\begin{pmatrix} 3 & 1 \\ -1 & 2 \\ \end{pmatrix}.$$ Let $V =\mathbb Z^2$ and $L = AV$. We want to find basis for $V$ and $L$ and ...
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### How to describe a ring after adjoining an element to it?

I want to know how to go about describing a ring after adjoining an element that satisfies a certain relation. As an example, I'm considering the ring obtained from Z3 by adjoining an element a ...
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### How to find the GCD of $29-3i$ and $11+10i$

I'm trying to find the GCD of $29-3i$ and $11+10i$ in $Z[i]$ I know that one way I can do this, is to factorize each of the numbers and find the GCD through that. But the norms in these two numbers ...
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### How to describe all ideals in a quotient ring?

$\newcommand{\C}{\mathbb{C}}$I'm a bit confused by quotient rings, and I want to know the general technique for describing their ideals. For example, say I have the ring $\C[x]/(x^3)$ If we consider ...
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### Is a p-Sylow subgroup of the underlying additive group of a finite commutative ring an ideal?

Let $R$ be a finite commutative ring. Let $H$ be a $p$-Sylow subgroup of the additive group $(R,+)$. Is $H$ an ideal in $R$? For ex: R= Z/6Z is a finite commutative ring. H={0,3} is a 2-sylow ...
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### Any hint with the next theorem?

"Let $GR=(s,sm)$" the Galois Ring with characteristic $p^s$ and $p^{sm}$ elements. If $n\mid m$ then exists $R_0=GR(s,sn)$ a Galois Ring that contains $R$ as a subring". I've already seen that this ...
1answer
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### Why $\mathbb{Z}_p$ over $\mathbb{Z}_{p^2}$ cannot be free?

Makoto Kato in the answer to this question wrote: Let $p$ be a prime number. Let $R = \mathbb{Z}/p^2\mathbb{Z}$. Let $M = R/pR$. Since the number of elements of $M$ is $p$, $M$ cannot be free....
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### Proving every nonzero element in a finite ring is either a unit or a zero divisor

I've just started learning ring theory, and the book I'm using uses "Every nonzero element in a finite ring is either a unit or a zero divisor" implicitly without explanation. So I came across this ...