# 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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### Every ideal of a finite principal ring is generated by a zero divisor

I stumbled upon this while doing introductory exercises to abstract algebra, but since i am still rather inexperienced in the subject, i would appreciate a second opinion. We will assume the ring to ...
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1 vote
1 answer
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• 1,369
13 votes
5 answers
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2 votes
3 answers
398 views

### $x^2+3x+3$ is irreducible in $\mathbb{F}_{25}[x]$

Give an example of an irreducible non-linear polynomial in $\mathbb{F}_{25}[x]$. I know that $x^2+3x+3$ is irreducible in $\mathbb{F}_{25}[x]$ but I know no shorter proof then the exhaustive search (...
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0 votes
2 answers
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• 1,174
4 votes
1 answer
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### Finite Non Commutative Rings of Cardinality n

For any given $n\in N$, Can we find a non-commutative ring of $n$ elements (with or without identity)? If not, can we find some condition on $n$ such that a non-commutative ring of $n$ elements ...
• 1,318
3 votes
3 answers
147 views

### How many solutions of $x^{p+1} \equiv 1 \mod p^{2017}$

How many solutions does $x^{p+1} \equiv 1 \mod p^{2017}$ have in set $\left\{0,1,...,p^{2017}-1 \right\}$? $p$ is prime > 2. My observations $1$ is one of solutions of given equation. $p$ is prime ...
0 votes
2 answers
683 views

• 594
-2 votes
2 answers
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### How to show that $2$ is invertible in a ring with odd cardinality?

Let $R$ be a commutative ring with unity that has an odd number of elements. Show that $2$ is invertible in $R$. Attempt I've found that $2 \ne 0$ from Lagrange, since the order of $1$ in the ...
• 1,351
5 votes
2 answers
69 views

### The number of polynomial functions $f:A\to A$ is $|A|^2$ if and only if $x^2=x$ for all $x\in A$.

Let $A$ be a commutative ring with $n$ elements, $n\ge2$. Prove that the next statements are equivalent: $(\forall x\in A)(x^2=x)$. The number of polynomial functions $f:A\to A$ is $n^2$. I managed ...
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0 votes
1 answer
60 views

### Is $\frac{R[x]}{(f(x))}$ finite for $R$ finite and $f$ not monic polynomial?

If $R$ is a finite commutative ring, then is $\frac{R[x]}{(f(x))}$ finite with $f$ not monic polynomial? I can prove above claim if f(x) is monic polynomial using division algorithm? But I am not ...
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-1 votes
2 answers
390 views

### How to solve systems of linear equations over a finite ring [closed]

I don't know where to start and how to go forth when solving system of equations in for example $\mathbb{Z}_{11}$. I have 2 different systems I want help with with a walkthrough to understand what is ...
1 vote
4 answers
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• 3,118
2 votes
2 answers
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### What is the name of $(\mathbb{Z}_2^s, \oplus, \odot)$ and where is it studied?

I'm studying the ring $(\mathbb{Z}_2^s, \oplus, \odot)$, where $s$ is arbitrary, $\oplus$ is the sum modulo $2$, and $\odot$ is the AND. Does it have a name? Even for a certain fixed $s>1$? Does ...
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4 votes
0 answers
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### Subring of $\text{Mat}_n(Z_m)$ is commutative if $x^2=0 \implies x=0$.

Let $A$ be subring of $\text{Mat}_n(Z_m)$. Suppose, for $x\in A$, $x^2=0$ implies $x=0$. Claim A is commutative. Attempt $A$ is finite ring, hence Artinian. If it is possible to claim that Jacobson ...
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0 votes
1 answer
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### How do I find a ring with a primary ideal having n elements?

I would like to know how can I find a ring with (at least) a primary ideal which has n elements (not generators, but elements) for a given n ? Thank you.
5 votes
1 answer
242 views

### Galois Theory for Finite Local Commutative Rings

Let $R\subseteq S$ be two finite commutative local rings with unique maximal ideals $m$ and $M$, respectively. We say that $S$ is a separable extension of $R$ if $mS=M$. We also say that $S$ is a ...
5 votes
2 answers
453 views

### Minimal ideal in commutative finite rings

Let $R$ be a commutative finite ring with identity, and let $I$ be a minimal ideal of $R$, that is, a non-zero ideal that there is no ideal strictly between $I$ and $0$. Now let $\{I_i\}_{i\in A}$ be ...
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