Mathematics

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.

222 questions
64
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
  • 723
50
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
  • 223,222
49
votes
6 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 by…
Freeman
  • 5,779
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
  • 6,403
32
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
  • 1,021
32
votes
5 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
  • 8,459
30
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
  • 2,264
23
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
  • 3,218
22
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
  • 27,289
22
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 unity…
Josh
  • 2,616
22
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
  • 2,261
19
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
  • 273
16
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
  • 16,701
16
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
16
votes
3 answers

Left inverse implies right inverse in a finite ring

Let $R$ be a finite ring with identity $1$ , and assume $\exists x,y\in R$ such that $ xy=1$ . How can I show it implies $yx=1$ ?
CC_Azusa
  • 1,553
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