Mathematics Stack Exchange
Mathematics Stack Exchange

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.

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

For an overview, see this Wikipedia entry.

212 questions
57
votes
5 answers

Every nonzero element in a finite ring is either a unit or a zero divisor

Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
rupa
  • 653
  • 1
  • 7
  • 11
44
votes
2 answers

Which finite groups are the group of units of some ring?

Motivated by this question: Which finite groups are the group of units of some ring? Which finite groups are the group of units of some finite ring? Which finite abelian groups are the group of units of some commutative ring? Which finite abelian…
lhf
  • 208,399
  • 15
  • 224
  • 525
34
votes
7 answers

Why is a finite integral domain always field?

This is how I'm approaching it: let $R$ be a finite integral domain and I'm trying to show every element in $R$ has an inverse: let $R-\{0\}=\{x_1,x_2,\ldots,x_k\}$, then as $R$ is closed under multiplication $\prod_{n=1}^k\ x_i=x_j$, therefore…
Freeman
  • 4,739
  • 11
  • 47
  • 75
33
votes
1 answer

Universal binary operation and finite fields (ring)

Take Boolean Algebra for instance, the underlying finite field/ring $0, 1, \{AND, OR\}$ is equivalent to $ 0, 1, \{NAND\} $ or $ 0, 1, \{ NOR \}$ where NAND and NOR are considered as universal gates. Does this property, that AND ('multiplication')…
Dilawar
  • 5,747
  • 6
  • 25
  • 36
29
votes
1 answer

A ring with few invertible elements

Let $A$ be a ring with $0 \neq 1 $, which has $2^n-1$ invertible elements and less non-invertible elements. Prove that $A$ is a field.
Claudiu Mindrila
  • 931
  • 5
  • 11
27
votes
3 answers

A finite ring is a field if its units $\cup\ \{0\}$ comprise a field of characteristic $\ne 2$

Suppose $R$ is a finite ring (commutative ring with $1$) of characteristic $3$ and suppose that for every unit $u \in R\:,\ 1+u\ $ is also a unit or $0$. We need to show that $R$ is a field. Is this true if ${\rm char}(R) > 3$? Here is what I…
algebra_fan
  • 2,154
  • 1
  • 15
  • 14
25
votes
4 answers

Ring of order $p^2$ is commutative.

I would like to show that ring of order $p^2$ is commutative. Taking $G=(R, +)$ as group, we have two possible isomorphism classes $\mathbb Z /p^2\mathbb Z$ and $\mathbb Z/ p\mathbb Z \times \mathbb Z /p\mathbb Z$. Since characterstic must divide…
Theorem
  • 7,367
  • 7
  • 52
  • 93
21
votes
2 answers

Structure of Finite Commutative Rings

Is every finite commutative ring $A$ a direct product of finite algebras over $\mathbb Z/p^n$?
zacarias
  • 3,118
  • 1
  • 14
  • 28
21
votes
2 answers

Prove that prime ideals of a finite ring are maximal

Let $R$ be a finite commutative unitary ring. How to prove that each prime ideal of $R$ is maximal?
Alex
  • 2,111
  • 16
  • 25
19
votes
4 answers

Is the group of units of a finite ring cyclic?

The group of units of a finite field is cyclic. Is it true that the group of units of a finite ring is also cyclic? If not, where does the ring structure prevents us from obtaining the result that is true for fields?
Manos
  • 24,205
  • 4
  • 51
  • 145
19
votes
6 answers

Does a finite commutative ring necessarily have a unity?

Does a finite commutative ring necessarily have a unity? I ask because of the following theorem given in my lecture notes: Theorem. In a finite commutative ring every non-zero-divisor is a unit. If it had said "finite commutative ring with…
Josh Chen
  • 2,489
  • 17
  • 31
15
votes
3 answers

How to show that a finite commutative ring without zero divisors is a field?

$R$ is finite commutative ring without zero divisors which has at least two elements. How to show that $R$ is field? I'm just starting with abstract algebra and I'd really appreciate if someone could explain it to me.
user62136
  • 253
  • 1
  • 4
  • 10
14
votes
3 answers

A finite commutative ring with the property that every element can be written as product of two elements is unital

I was struggling for days with this nice problem: Let $A$ be a finite commutative ring such that every element of $A$ can be written as product of two elements of $A$. Show that $A$ has a multiplicative unit element. I need a hint for this…
mathfan
  • 491
  • 2
  • 9
14
votes
4 answers

Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.

Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime. My attempt: Let $I$ be an ideal of $R$. Then we have $I$ is maximal $\Leftrightarrow$ $R/I$ is a finite field $\Leftrightarrow$ $R/I$ is a finite…
Idonknow
  • 14,077
  • 20
  • 77
  • 208
14
votes
5 answers

Finite quotient ring of $\mathbb Z[X]$

Since userxxxxx (I don't remember the numbers) deleted his own question which I find interesting, let me repost it: Let $f,g\in\mathbb Z[X]$ with $\mathrm{gcd}(f,g)=1$. Prove that the ring $\mathbb Z[X]/(f,g)$ is finite.
user26857
1
2 3
…
14 15