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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1answer
18 views

Invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$

If $A$ is a ring with unit element $1 \ne 0$ let $A^{\times}=\{a \in A: a$ invertible$\}$. Find the invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$. ...
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1answer
35 views

Calculation of the unit group of a finite ring

Is there an easy/fast way using GAP to calculate the unit group of a finite ring? For example, the Units command does not work for some finite rings: (I'm using the ...
1
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1answer
29 views

Ireducible polynomial over $\mathbb Z_4$

How can you prove that $f(x)=X^2+1$ is ireductible over $\mathbb Z_4$, the quotient ring? We know that $\mathbb Z_4$ admits divisors of $0$, as $2*2=0$, so any elemanary approach using $h\times g=f$ ...
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2answers
50 views

Number of elements in $S^{-1}(\mathbb Z/6\mathbb Z)$ [closed]

I'm stuck in the following computation : Given $S=\{6\mathbb Z+1,6\mathbb Z+3,6\mathbb Z+5\}$, we define $S^{-1}(\mathbb Z/6\mathbb Z)=(\mathbb Z/6\mathbb Z)/S$ ...
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1answer
44 views

A finite commutative ring with at least two elements consisting of no zero divisors is a field

My question was that finite ring of non divisors elements forms a field ... My approach to this was let all the elements be 0 , x1,x2 ,...xn then x1.x2.x3.....xn = xj for some j <=n then xj.( ...
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2answers
58 views

Properties of finite ring

Suppose $R$ is a finite ring. It may be commutative or it may be not. Let $x \in R$. Show that there exists a positive integer $n$ such that $x^n = x^{2n}$. Is it true that $x^{k!} = x^{2k!}$, where ...
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2answers
42 views

On analogy between $\Bbb Z$ and $\Bbb F_q[x]$

There are objects and operations analogous between $\Bbb Z$ and $\Bbb F_q[x]$. For example primes in $\Bbb Z$ and irreducibles in $\Bbb F_q[x]$ are analogous and so is multiplication operation. ...
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1answer
41 views

Is $\Bbb Z / p$ where $p$ is not prime a PID? [closed]

Is it possible for a finite ring with unity in the form of $\Bbb Z / p$ where $p$ is not prime to be a PID?
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0answers
38 views

If $k \equiv ab \pmod n$ then $k^{-1} \equiv a^{-1} b^{-1} \pmod n$

How can I prove that if: $k \equiv ab \pmod n$ then $k^{-1} \equiv a^{-1} b^{-1} \pmod n$ ? Here is my attempt: Start from this congruence: $K K^{-1} \equiv 1 \pmod N$ Then try to reach this ...
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3answers
87 views

How many zero divisors are in a finite ring?

I'd like to know how many zero divisors there are in the ring $\mathbb{Z}_5[x]/(x^3-2).$ Is it sufficient to note that for $p(x) = x^3-2,$ $p(3) = 0$ and $p(5)=0?$ I haven't been able to write ...
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1answer
57 views

Finite integral domain

I encountered a problem: Every finite integral domain is isomorphic to $ \mathbb{ Z }_{p} $. I know that finite integral domain is isomorphic to a field, but I have no idea on how to construct ...
2
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1answer
60 views

A finite ring without zero divisors has a neutral element for the multiplication.

$(R,+,\cdot)$ is a finite ring without zero divisors. Prove that $R$ has a neutral element for the multiplication. Can someone give a hint or something? Thanks.
3
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1answer
122 views

A finite von Neumann regular ring is unital and has $ab = 1$ if $ba = 1$

Let $R$ be a finite ring satisfying for any $x \in R$ there exists $y \in R$ with $xyx = x$. Show that $R$ is unital and that if $ab = 1$, then $ba = 1$. Thoughts so far: If I can show that the ...
3
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3answers
120 views

How many elements does this ring have?

I know that the following ring is not a field because the defining polynomial is reducible into two polynomials that are irreducible: $$\mathbb Z_2[X]/(x^5+x+1)$$ where $$x^5 + x + 1 = (x^2 + x + 1) ...
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0answers
99 views

Zero-Sum Partitions of Nonzero Elements of a Ring

In this question, rings are not necessarily finite nor do they need to be unital (i.e., the multiplicative identity may not exist). Although I shall almost exclusively discuss finite commutative ...
3
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1answer
76 views

Finding all ideals in a finite ring

Let $\mathbb F_2$ be the field of two elements. Consider the factor ring $$R=\mathbb F_2[x, y]/\langle x^2, y^2\rangle.$$ I want to find all ideals of $R$. Note that ...
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1answer
50 views

Ring Structure for Non Commutative Groups: Is there a grander reason for Abelian requirements?

So I was considering the following idea. Let a generalized Ring $gR$ be a set equipped with two operations $$u_1, u_2$$ Such that $gR$ is a with the operation $u_1$ and the operation $u_2$ ...
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0answers
106 views

Smallest Two-Sided Nearring

For those who are unfamilar with nearrings, here is a definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both left ...
0
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1answer
80 views

Finding all homomorphisms between rings

I am looking for a good method to understand how to find all possible homomorphisms between rings, e.g $\varphi :\mathbb{Z}\rightarrow \mathbb{Z}$ or, as another example: $\varphi ...
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1answer
85 views

Find all subrings of a ring

Given a finite ring, e.g $\mathbb{Z}{_{24}}$, how can I find all of its subrings? I have tried to think about it couldn't reach any idea. Thanks.
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1answer
43 views

Equations in local rings

Let $R$ be a finite commutative local ring with identity. Assume that every ideal in $R$ is principal. Let $u$ and $v$ be units in $R$ and let $z\neq 0$ be a zero divisor. I think that $uz=vz$ ...
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1answer
25 views

Find an ideal of $R = \mathbf{Z}/12\mathbf{Z} = \{0,1,2,\dots,11\}$ with two elements

Let $R = \mathbf{Z}/12\mathbf{Z} = \{0,1,2,\dots,11\}$. Find an ideal $I$ of $R$ which consists of two elements. How many elements does $R/I$ have? I thought the ideals would be $\{0\}$, ...
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2answers
106 views

Is there a finite ring whose rank is smaller than the rank of its group and its monoid?

Consider a finite ring $(R, +, \times)$ comprising a finite additive abelian group $(R, +)$, a finite multiplicative monoid $(R, \times)$, and a distributivity rule relating the two. Let the rank of ...
3
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1answer
83 views

Algebraic closure for rings

Is there any notion of algebraic closure for commutative rings? I am specifically interested in such a concept for $\mathbb Z_n$, with $n$ not a prime (possibly square-free). Such a concept would be ...
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2answers
58 views

a question about abstract algebra,the order of $\Bbb Z_{5}[x]/ (x^3+x+1)$

Firstly, I have proven that $x^3+x+1$ is irreducible in $\Bbb Z_{5}[x]$,then how can I know the order of $$\Bbb Z_{5}[x]/ (x^3+x+1),$$ where $(x^3+x+1)$ means the ideal generated by $x^3+x+1$. Can ...
5
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3answers
156 views

Finite commutative ring with more than $\frac{2}{3}$ of its elements idempotent

Suppose that $R$ is a finite commutative ring with identity element, such that more than $\frac{2}{3}$ of elements are idempotent. Prove that all of elements are idempotent. Please give me a ...
2
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1answer
111 views

Characterize all finite unital rings with only zero divisors

Is it true that for every finite (for simplicity, commutative) ring $R$ in which every element not equal to $1$ is a zero divisor, is isomorphic to the zero ring or $\mathbb{Z}/2\mathbb{Z}$, ...
2
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1answer
57 views

Algebraic and transcendental functions over $\mathbb Z_n$ — is this a known result?

I have proved a result that seems (to me) interesting, and I am wondering whether it is a known result, and if not whether it seems interesting to others. The result is as follows: Let $R=\mathbb Z ...
2
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1answer
89 views

Find a ring homomorphism $\tau: \mathbb{F} \rightarrow \mathbb{F}$

Just working on some exam prep questions, and I'm a bit stuck on this one. Let $ \mathbb{F} = \{ a + bX + cX^2 | a,b,c \in \mathbb{F}_2 = \{0,1\} \} $ be a ring with the operations: Addition, ...
3
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2answers
63 views

Every finite ring with identity $a+a = 0$ is subring of $Mat_{n\times n} (\mathbb{F}_2)$ for some $n$?

Is it true, that every finite ring $R$ with identity $\forall a \in R (a+a=0)$ is subring of $Mat_{n \times n}(\mathbb{F}_2)$ for some $n$?
2
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0answers
55 views

On units in subrings and quotient rings of a finite ring with unity

Let $R$ be a finite ring with unity and $S$ be an ideal (or subring). Let $R^*$ be the group of units of $R$ and $S^*:=R^* \cap S$. Then does $|S^*|$ divide $|R^*|$ ? Moreover, if $I$ is an ideal of ...
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0answers
43 views

Unfamiliar w/ Ring Notation

I'm used to seeing rings represented as sets, but in one of my homework problems, I am asked to: Find the number of zero-divisors of $R_{x^2-x}$. Can somebody please explain what this notation ...
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1answer
56 views

How to find all ring structures over $C_2\times C_2$?

$C_2$ denotes the cyclic group of order 2. How to find all ring structures over $C_2\times C_2$? The question is equivalent to give a full list of all essentially different bilinear 2-operations ...
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2answers
57 views

The cardinality of $\text{GL}(n, \mathbb{Z} / m\mathbb{Z})$

Given a ring $R$, let $\text{GL}(n, R)$ be the group of invertible $n \times n$ matrices with entries in $R$. I know the easy counting argument that shows that $\text{GL}(n, \mathbb{F}_q)$ has $(q^n ...
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2answers
64 views

A finite pseudo-ring such that $ab^2=b$ for some $(a,b)\in A$

I have this exercise which I found very difficult: $(A,+,.)$ is a finite pseudo-ring (a structure satisfying all the axioms of a ring except for the existence of a multiplicative identity), and ...
3
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2answers
225 views

Ring with many one-sided zero-divisors

Does there exist a ring all of whose elements are left zero-divisors but only one element is a right zero-divisor? The motivation for asking this question is that if there exists atleast one left ...
2
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1answer
119 views

the number of zero divisors in polynomial ring

I was looking for an answer on the question How much zero divisors are in the ring $\dfrac{\mathbb{Z}_3[x]}{(x^4 + 2)}$? when I came up with the brilliant/hack-isch idea that it might just be ...
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1answer
222 views

There are only two non isomorphic rings with $p$ elements

Prove that for any prime $p$ there are only two non isomorphic rings with $p$ elements. I have found out there are up to two rings of order p , they are $\mathbb Z_p$ and $\mathbb C_p$. Please ...
2
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0answers
68 views

Finite Ring with unity and no zero divisors is field [duplicate]

I would like to know if someone can help me with this. "Show that a finite ring $R$ with unity $1\neq 0$ and no divisors of 0 is a field." The original exercise asked me to show that it was a ...
5
votes
1answer
142 views

Example of finite ring with a non principal ideal

I would like to have an example (or a class of examples) of a finite ring (with unit, not necessarily commutative) that is not a principal ideal ring (if possible an example of a local ring and of ...
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1answer
264 views

Any ring of prime order commutative ?

Is any ring of prime order commutative ?
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1answer
41 views

How to verify an algebraic structure is a ring

I have a problem which ask me to verify that to structures are rings. However, I'm unsure of how exactly to check each property. I believe that the first is closed but not sure how to check the ...
5
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1answer
95 views

Units in a finite ring

Let $A$ be a finite unital ring and let $N$ be the set of nonunits of $A$. I want to show that if $|N|>1$ then $\sqrt{|A|}\leq |N|$. I have tried to find an injective function from $A$ to $N$, ...
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0answers
92 views

How many unique combinations of sets can we get?

Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists. Set $p$ a prime greater than 2, $\alpha = (p-1)/2$, and ...
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1answer
51 views

Is every finite ring a matrix algebra over a commutative ring?

In this MO answer it is stated that every finite ring is a direct sum of finite-dimensional algebras over $\mathbb{Z}/p^k$ for varying $p$ and $k.$ What I am wondering is the following: Can every ...
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1answer
79 views

Is this “sliding window” unique?

Starting with $x$, which is a positive integer or zero, and $y$ a second positive integer or zero, with $y \ne x$, we can create lists. Set $p$ a prime greater than 2, $\alpha = \lfloor p/2 \rfloor$, ...
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0answers
62 views

A finite ring with $p^3$ elements satisfying some condition is local

Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p{^3}$ where $p$ is a prime number. Prove that if the number of elements of $Z(R)$ (the set of zero ...
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0answers
43 views

finding all rings of order $p^3$

Classify all commutative rings with $1$ of order $p^3$. I only observed $1$ can have order $p,p^2,p^3$. If its $p^3$ then $\mathbb{Z}/(p^3)$. Otherwise will it be just be $\mathbb{Z}/(p^2)\times ...
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3answers
109 views

Let $A$ a finite ring. Prove that $A$ is a field, or $A$ has zero divisors. [duplicate]

Let $A$ a finite ring. Prove that $A$ is a field, or $A$ has zero divisors. I begin to assume that $A$ has no zero divisors but I don't know continue... \ How would be this proof? thanks! :)
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0answers
38 views

Finite Division Rings are Fields [duplicate]

I have seen a problem recently. It says that every finite division ring is a field. How to show this?