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 ...
paulina's user avatar
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1 vote
1 answer
175 views

Linear congruences over a finite ring are equivalent

Let $n,N$ be two natural numbers. Let $a=(a_1, ..., a_n),b=(b_1, ..., b_n)\in (\mathbb{Z}/N\mathbb{Z})^n$ with $\gcd(a_1, ..., a_n, N) =1$ and $\gcd(b_1, ..., b_n, N) =1$. Define $H_a = \{(x_1,...,x_n)...
user avatar
2 votes
0 answers
120 views

Subrings and quotients of finite semigroup algebras

Let $F = \Bbb F_p$ be a prime finite field and $R$ an arbitrary finite-dimensional associative (+ let's say unital) algebra over $F$. Then $R$ is a subalgebra (=subring here) of a matrix algebra $M_n(...
Amateur_Algebraist's user avatar
13 votes
5 answers
3k views

Why does Gaussian elimination sometimes work in rings where it should not?

I think it's best to illustrate this with an example. Take for instance the ring of integers modulo $6$. If I have the system of equations: $$ \begin{aligned} 2x + 2y &= 4 \\ 3x + 4y &= 3 \end{...
james Orr's user avatar
  • 141
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0 answers
31 views

Lifting Idempotents in McDonald's Finite Rings w/ Identity Theorem XIV.3

Assume that $S$ and $R$ are finite commutative local rings with maximal ideals $M$ and $m$ respectively. I am working through the proof of Theorem XIV.3 in McDonald's Finite Rings with Identity The ...
dbossaller's user avatar
4 votes
1 answer
113 views

A polynomial with unique root in every $ \mathbb Z _ n $

Let $ p ( x ) \in \mathbb N [ x ] $ be a polynomial with nonnegative integer coefficients, and $ a \in \mathbb Z $ be a given integer constant. If for all positive integers $ n $, $ p ( x ) + a $ has ...
Mohsen Shahriari's user avatar
2 votes
1 answer
568 views

Classification of quadratic forms over $\mathbb{Z}/n\mathbb{Z}$ - even characteristic case

Let $R$ be a ring (unital, commutative) and $M$ a free $R$-module of finite rank. A quadratic form is a map $q:M\rightarrow R$ such that $\forall r\in R:\forall m\in M: q(rm)=r^2\cdot q(m)$ and the ...
Thomas Preu's user avatar
  • 2,002
4 votes
2 answers
1k views

Show that no ring of order 6 is an integral domain.

Problem Statement: Show that no ring of order $6$ is an integral domain. Some Definitions: Integral Domain: a commutative ring with identity and no zero divisors. $\mathbb{Z}_n$: group of the elements ...
Off Kilter's user avatar
3 votes
1 answer
152 views

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 ...
worldsmithhelper's user avatar
2 votes
2 answers
157 views

Diagonalizable binary matrices over $\mathbb{Z}_4$

I'm trying to figure out the following question: Are symmetric, binary $n\times n$ matrices with zeros on the diagonal, are diagonalizable over $\mathbb{Z}_4$? I know that it isn't true that $\textit{...
GWB's user avatar
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2 votes
0 answers
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Finite ring with irreducible element that is not prime

Let $R$ be a commutative unital ring. Let's call a non-zero non-unit $a \in R$ irreducible if $a=bc$ implies that (either) $b$ or $c$ is a unit, prime if $a \mid bc$ implies that $a \mid b$ or $a \mid ...
Thrash's user avatar
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5 votes
1 answer
401 views

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 ...
Charlotte's user avatar
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4 votes
1 answer
216 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 ...
frafour's user avatar
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2 votes
1 answer
216 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 $j$...
NoDisplayName's user avatar
3 votes
1 answer
118 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 ...
frafour's user avatar
  • 3,015
9 votes
2 answers
358 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-...
BlueSyrup's user avatar
  • 235
3 votes
1 answer
134 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 ...
GreginGre's user avatar
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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^...
maryam's user avatar
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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 (...
Maxim Nikitin's user avatar
0 votes
2 answers
19 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)>} \...
Gustavo Terra's user avatar
1 vote
1 answer
347 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 ...
UnsinkableSam's user avatar
3 votes
1 answer
57 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 ...
Pouya Layeghi's user avatar
0 votes
1 answer
395 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 ...
S.S's user avatar
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1 answer
526 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 $S$ ...
silver's user avatar
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1 vote
1 answer
55 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 ...
A.A.'s user avatar
  • 361
1 vote
1 answer
85 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)....
mrde05's user avatar
  • 153
2 votes
0 answers
38 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 ...
Peter Kagey's user avatar
  • 5,052
1 vote
1 answer
29 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 ...
Paweł Piwek's user avatar
2 votes
1 answer
57 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\...
Zongxiang Yi's user avatar
  • 1,174
4 votes
1 answer
461 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 ...
SARTHAK GUPTA's user avatar
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 ...
user avatar
0 votes
2 answers
683 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+\...
Flo's user avatar
  • 113
4 votes
2 answers
53 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 ...
Matthew Hannigan's user avatar
4 votes
2 answers
421 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 ...
magikarrrp's user avatar
2 votes
2 answers
362 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 ...
Marc's user avatar
  • 1,208
1 vote
2 answers
68 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 $\...
MNM's user avatar
  • 594
-2 votes
2 answers
262 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 ...
tyuiop's user avatar
  • 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 ...
tyuiop's user avatar
  • 1,351
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 ...
Curious student's user avatar
-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 ...
user633788's user avatar
1 vote
4 answers
2k 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,[...
yLccc's user avatar
  • 417
5 votes
0 answers
130 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 $R/I$ ...
Bib-lost's user avatar
  • 3,900
3 votes
0 answers
98 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 $\phi\big(f(x)\big)=f(i)$. ...
fivestar's user avatar
  • 919
2 votes
3 answers
662 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 ...
jasmine's user avatar
  • 14.5k
2 votes
1 answer
42 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 ...
Conjecture's user avatar
  • 3,118
2 votes
2 answers
81 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 ...
Guillermo Mosse's user avatar
4 votes
0 answers
130 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 ...
Jo''s user avatar
  • 548
0 votes
1 answer
44 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.
Ruxandra Mihaela Ichim's user avatar
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 ...
J.Johnson's user avatar
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 ...
Deroty's user avatar
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