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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4
votes
1answer
60 views

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 ...
2
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1answer
53 views

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 $j$...
2
votes
1answer
41 views

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 ...
4
votes
1answer
82 views

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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0answers
24 views

All roots of a polynomial in ring $F_2+uF_2+u^2F_2$, where $u^3=0$

Let $R=F_2+uF_2+u^2F_2$, where $u^3=0$, be a finite commutative ring. So $R=\{0,1,u,v,uv,u^2,v^2,v^3\}$, where $v=1+u$, $v^2=1+u^2$, $v^3=1+u+u^2$, $uv=u+u^2$. It is well known that $$x^7-1=(x+v^3)(x^...
2
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3answers
93 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 (...
0
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2answers
17 views

Is it possible rewrite $\frac{\mathbb{Z}_m [x]}{<f(x)>}$ as a in direct sum for $m$ composite?

I have a doubt: Given $\frac{\mathbb{Z}_m [x]}{<f(x)>}$, where $m$ is composite and $f(x)=\prod_{i=1}^t f_i^{a_i} (x)$ (irreducible factors), can I admit $\frac{\mathbb{Z}_m [x]}{<f(x)>} \...
1
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1answer
48 views

Is a ring with order $2$ unique? (In the sense of isomorphism)

Apologies. My definition of a ring need not include the multiplicative identity. I had this problem asking if there exists rings $R_1, R_2$ such that they both have order $2$ but they are not ...
3
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1answer
42 views

Prime ideals of $F_q[C_m]$

My main question is that what the prime ideals of group ring $F_q[C_m]$ are where $F_q$ is a finite field with $q$ elements and $C_m$ is a cyclic group of order m. To do this, I was thinking that how ...
0
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1answer
72 views

Existence of $n$ such that $a^n=a$ for all $a$ in $Z_m$ [duplicate]

This is a question from Contemporary Abstract Algebra which asks: Find an integer $n > 1$ such that $a^n = a$ for all $a$ in $Z_6$. Show that no such $n$ exists for $Z_m$ when $m$ is divisible ...
1
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1answer
93 views

Ring homomorphism from matrix ring to smaller ring

Let $\mathbb{F}$ be some finite field, and let $R := M_n(\mathbb{F})$ be the set of $n$-by-$n$ matrices over $\mathbb{F}$. Then $R$ is finite. Does there exist some pair $(\varphi, S)$ such that $S$ ...
1
vote
1answer
47 views

Ultimately show $R\cong S$ implies $R[x]\cong S[x]$.

The question above is going to be used to ultimately show that $R\cong S$ implies $R[x]\cong S[x]$. I understand how these results imply that, but I am having trouble actually doing the proof parts ...
1
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1answer
52 views

Constructing the group SL$_2$ of $\mathbb{Z}[i]$ mod $p$ in GAP

I'm trying to construct, in GAP, the group $$\mbox{SL}_2(\mathbb{Z}[i]/p),$$ where $p$ is a prime in $\mathbb{Z}[i]$, the Gaussian integers. $p$ could be real or complex (for $p$ real would be enough)....
2
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0answers
28 views

Show finite ring has identity when for each $x$ there exists $y$ such that $xyx = x$. [duplicate]

I'm working through an old qualifying exam problem, and I'm stuck: Let $R \neq (0)$ be a finite ring such that for any element $x \in R$ there is $y \in R$ with $xyx = x$. Show that $R$ contains an ...
1
vote
1answer
23 views

GAP — GeneratorsOfRing giving a list with repeated element

I put the following code in GAP: R := Integers mod 8; and I get the answer: [ ZmodnZObj( 1, 8 ), ZmodnZObj( 1, 8 ) ] Can ...
2
votes
1answer
49 views

Find finite rings $(R,+,\times)$ such that for every unit $r$, $r-1$ is a unit except $r=1$.

Let $(R,+,\times)$ be a finite ring. $R^\times$ denotes the set of all invertible elements, i.e., units in $(R,\times)$. Find finite rings $(R,+,\times)$ such that for every unit $r\in R^\times\...
4
votes
1answer
138 views

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 ...
3
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3answers
83 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
2answers
194 views

Ideals and order of a polynomial ring

Consider the ideal $I=(X^3+\hat2X+\hat1)$ of polynomial ring $R=\mathbb Z_3[X]$. Is $R/I$ an integral domain? How many elements does $R/I$ have? Find the inverse of $X^3+\hat1$ in $R/I$. (1) $(X^3+\...
4
votes
2answers
49 views

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 ...
4
votes
2answers
301 views

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 ...
2
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2answers
112 views

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 ...
1
vote
2answers
46 views

Finding inverses in quotient rings

In $ A=\mathbb{Z}[i]=\{a+bi \ : \ a,b \in \mathbb{Z}\} $ we consider $a=7+56i; \ b=3+3i; \ c=1+8i$. We will write $(a)$ to refer to the ideal generated by $a$ Find out whether the elements $\...
-2
votes
2answers
127 views

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 ...
5
votes
2answers
59 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 ...
0
votes
1answer
52 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 ...
-1
votes
2answers
141 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
4answers
323 views

How to prove $\Bbb Z[i]/(1+2i)\cong \Bbb Z_5$? [duplicate]

How to prove $\Bbb Z[i]/(1+2i)\cong \Bbb Z_5$? My method is $$(1+2i)=\big\{a+bi丨a+2b≡0\pmod 5\big\},$$ So any $a+bi$ in $\Bbb Z(i)$,we got $$a+bi=(b-2a)i+a(1+2i).$$ So $\Bbb Z[i]/(1+2i)=\big\{0,[...
5
votes
0answers
95 views

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 $R/I$ ...
3
votes
0answers
88 views

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 $\phi\big(f(x)\big)=f(i)$. ...
2
votes
3answers
176 views

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 ...
2
votes
1answer
21 views

if a ring is finite then the translation $x\rightarrow ax$ is surjective where $a\in A$ is regular

In a proof of the inversibility of regular elements in a finite ring, there is the following argument: let $A$ be a finite ring and $a\in A$ regular . the translation $A\rightarrow A: x\rightarrow ...
2
votes
2answers
78 views

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 ...
4
votes
0answers
117 views

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 ...
0
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1answer
39 views

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.
-1
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1answer
36 views

Minimal Polynomials over an extension field [closed]

Is there any easy way to solve this? Find the minimal polynomial of $(\sqrt{2})$+$(\sqrt[5]{3})$ over $\mathbb{Q}$.
5
votes
1answer
156 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
2answers
349 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 ...
2
votes
1answer
105 views

Identifying the ring $R=\mathbb{Z}_{9}[x]/(x^2-3,3x)$

I want to know if there is a simple form of the ring $$R=\mathbb{Z}_{9}[x]/(x^2-3,3x)$$ I tried to start with the equations $3x\equiv0$ and $x^2\equiv 3$. So, $3x^2\equiv 0$ and $3x^2\equiv 9$. ...
5
votes
1answer
109 views

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 ...
0
votes
1answer
64 views

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 $GF(2)[x]/p(x)$ ...
0
votes
1answer
138 views

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 $\mathbb Z_2$...
2
votes
2answers
226 views

On polynomials over finite ring

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 \...
2
votes
1answer
64 views

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, ...
2
votes
1answer
85 views

Extension of isomorphism of submodules

If $V$ is an $n$-dimensional vector space over $\mathbb{F}_q$, and $V_1$ and $V_2$ are subspaces of $V$ (both k-dimensional), there exists an isomorphism $L: V \to V$, where if $u \in V_1$, then $L(u) ...
5
votes
1answer
137 views

Zero divisors on finite rings

I'm trying to prove or disprove the following fact: Let $R$ be a finite ring and $x \in R$ be a zero divisor. If $x$ is not nil-potent, then for some $n > 1$ we have $x^n = x$, i.e., $x$ is $n$-...
0
votes
1answer
85 views

Why is $\mathbb{Z}_m$ a commutative ring?

I understand for it to be a ring it has to be closed under the addition and product, have inverses for the additive group, distributive and identity in the product. I can see that it is closed ...
3
votes
3answers
594 views

For a finite local ring $R$, why $|R|$ is a prime power. [closed]

When I read about finite local rings, every one firstly suppose that $|R|=p^t$ where $R$ is a finite local ring and $p$ is a prime. I don't understand why this can be supposed ?
0
votes
1answer
3k views

Finite integral domain is a field: proof condition

Theorem: a finite integral domain is a field proof: Let D be a finite integral domain with unity 1. Let a be any non-zero element of D. If a=1, a is its own inverse and the proof concludes. Suppose,...
0
votes
1answer
100 views

Find finite rings $(R, +, \times$) such that $(R,+)$ is cyclic.

I'm finding finite rings $(R, +, \times$) such that $(R,+)$ is cyclic. $\mathbb{Z}_n$ is a good example. Up to ring isomorphism, is there any exmaples other than $\mathbb{Z}_n$ ? Thanks. For ...

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