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
26 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 ...
0
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1answer
32 views

Any ring of prime order commutative ?

Is any ring of prime order commutative ?
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1answer
20 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 ...
3
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0answers
63 views

Units in a finite ring

Let $A$ be a finite ring and $N$ 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 $N$, but I don't ...
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0answers
40 views

Finding the group structure of a finite ring

Trying to construct an example I built up this finite ring: $$B=\mathbb{Z}/9\mathbb{Z}[x,y,z,w_1,w_2]/(x^3-1,y^3-1,(x-1)(z+3w_1),(y-1)(z+3w_2),w_1^2,w_2^2,z^2)$$ I need to know the structure of ...
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0answers
45 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 ...
2
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1answer
28 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
71 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
29 views

Classify rings of order $p^3$ [duplicate]

Classify unital rings of order $p^3$. For the simplest case of order 8. Attempt:In a unital ring $R$ of order $8$ the additive order of $1$ can only be $2,4$ or $8$. In the third case when additive ...
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0answers
24 views

Classifying Unital Rings of order 8

Classify unital rings of order 8. Attempt:In a unital ring $R$ of order $8$ the additive order of $1$ can only be $2,4$ or $8$. In the third case when additive order of $1$ is 8, $1$ generates the ...
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0answers
43 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
29 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
65 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
34 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?
2
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1answer
82 views

$R$ be a ring without identity. If $R$ has a maximal left ideal, then the Jacobson radical is still the intersection of all the maximal left ideal?

We know that the definition of the Jacobson radical $J(R)$ (a) in a ring $R$ with identity is: $$J(R)=\cap \mbox{ maximal left ideals}.$$ (b) in a ring $R$ without identity is: $$J(R)=\{a\in R\mid ...
0
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1answer
30 views

Finite rings without unity that are subrings of finite rings with unity

I know that a ring $R$ without unity can be embedded as a subrng of a ring with underlying additive structure $R \oplus \mathbb{Z}$, a ring with unity. But this does not yield a finite field. But I ...
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2answers
59 views

Why is $1 + 1 = 0$ in $\{0, 1\}$ (binary field) and not 1 or 2?

Stuck on the simplest case in my foray into fields... I know there is a really similar question out there, but I can't find any contradiction with the field axioms if 1 + 1 = 1 instead of 0. Can ...
0
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1answer
46 views

$R$ is a ring. $(R,+)\cong(Z_2\oplus Z_4,+)$, indecomposable, $\nexists$ 1, noncommutative, has an idempotent $e\neq 0$. Show that $2e\neq 0$.

Let $R$ be an indecomposable ring with $(R,+)\cong(\Bbb{Z}_2\oplus \Bbb{Z}_4,+)$. Suppose that $R$ is noncommutative and has no multiplicative identity. If $R$ has an nonzero idempotent element $e$, ...
3
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1answer
76 views

How to characterize all finite commutative local rings with the maximal ideal of fixed order (if the order is small)?

Let $R$ be a finite commutative local ring with the maximal ideal $M$ of order $m$. How to characterize all such finite commutative local rings? For examples, if $m=2$, then $R\cong\mathbb{Z}_4$ ...
2
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2answers
94 views

When a finite local ring $R$ has $-1$ as a square in $R^\times$?

Let $R$ be a finite local ring with maximal ideal $M$ such that $|R|/|M|\equiv 1\pmod{4}$. Then $-1$ is a square in $R^\times$ (that is, there exists $u\in R^\times$ such that $u^2=-1$) if and only ...
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2answers
50 views

Any finite ring is a direct sum of rings of prime power order

I read some books and articles like http://goo.gl/P4VWS1 or http://goo.gl/FFyRup, which state a theorem: Any finite ring is a direct sum of rings of prime power order. But they only state the theorem ...
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1answer
48 views

Structure Theorem for Finite Commutative Rings with unity [duplicate]

The Structure Theorem for Finite Commutative Rings with unity state that: A finite commutative ring $R$ with multiplication identity is isomorphic to a direct sum of local rings. Suppose all the ...
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2answers
53 views

Building quotient rings

The quotient rings are following: $\mathbb{Z}[i]/(1+i)$, $\mathbb{Z}[i]/(1+2i)$, $\mathbb{Z}[\sqrt{-2}]/(2)$, $\mathbb{Z}[\sqrt{-2}]/(1+ \sqrt{-2})$. I know that the two first are likely to be ...
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2answers
52 views

Number of non trivial ring homomorphisms from $Z_{12} \ \ to \ \ Z_{28}$

One homomorphism is $ 1 \mapsto 1$, Other homomorphisms are : We know that if $f$ is a homomorphism from $R$ to $S$ and $f(1_R) \neq 1_S$, then $f(1_R)$ is a zero diviasor in $S$. So zero divisor of ...
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0answers
14 views

Is there anything interesting we can do with this fact on iterates of polynomials over a finite ring?

Let $R$ be a finite ring and $r \in R$ such that $f \in R[X]$ is a min. poly of $r$ over $R$. Ie. $f(r) = 0$ and $f$ has minimal degree. In particular if $g(r) = 0$ and $g \in R[X]$, we have that $f ...
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2answers
83 views

Can anyone explain the difference between a ring, a group, and a field in a way so that your average 15 year old can understand? [duplicate]

I can not understand key difference between them, because all on the web uses mathematical notations...Can anyone explain in easy way?...please.Thanks in advance.
2
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1answer
68 views

Example of a finite ring with identity containing a ring without identity

What is an example of a finite ring $R$ with unity and a subring $S$ of $R$ that is not a ring with unity?
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1answer
98 views

Quotient Ring of finite order with root of irreducible polynomial

The question is as follows: Suppose that $\alpha\in\mathbb{C}$ is a root of the irreducible polynomial $f(t) = t^d + \sum_{i=0}^{d-1}a_it^i$ , where $a_i\in\mathbb{Z}$ ($0\le i\le d-1$). Let ...
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2answers
59 views

Subring of $\Bbb Z_{18}$ with unity

Need help finding subrings $A$ and $B$ of $\Bbb Z_{18}$ in which $A$ and $B$ are rings with unity, $B$ is a subring of $A$, but the unity of $B$ is not the same as the unity of $A$.
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3answers
233 views

How many elements are there in this quotient ring?

So we are having this undergraduate course in my department of commutative algebra and there is a problem sheet that we have to submit. The second problem goes like this: Let $R$ be the ring ...
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1answer
46 views

Is $x^2 + 1$ irreducible over $\mathbb{Z}/_{3}[x]$?

My Problem is to consider if the polynomial $x^2 + 1$ is irreducible over $\mathbb{Z}_{/3}[x]$ My Approach was: after looking closer onto the given Facts, i can see that $\mathbb{Z}/_{3}[x]$ is a ...
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1answer
39 views

How many zero divisors at $\mathbb{Z}_{80}\times\mathbb{Z}_{100}$?

How many zero divisors at $\mathbb{Z}_{80}\times\mathbb{Z}_{100}$? I assume it: $(80-\varphi(80))\cdot(100-\varphi(100))$, I'm right or I miss somthing?? Thank you!
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0answers
62 views

Prove that this factor ring is a finite ring without zero divisors [duplicate]

Let $R=\mathbb{Z}[\sqrt{d}]=\{a+b\sqrt{d}\mid a,b,d\in\mathbb{Z}\}$ and let $d$ be square-free. Let $P$ be a non-zero prime ideal in $R$. I need to prove that the factor ring $R/P$ has no zero ...
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1answer
43 views

Find a finite ring with elements other than the zero element, units, or zero-divisors.

I'm asking this because I could only think of infinite rings where this is true. This must include rings.
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2answers
43 views

*Step* in proving that there are infinitely many primes that suffice…

Let $k,n\in \mathbb Z$ with $n=k^2+1$ and let $p$ be an odd prime with $p\mid n$. Prove that $p\equiv1\text{ mod }4$. I found out that $\bar{n}\in\left\{ \bar{1},\bar{2}\right\} $ (denoting ...
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2answers
29 views

Non boolean example of a finite ring $R$ with $r^4 = r$ for all $r$ in $R$.

I just proved that a finite ring $R$ with $r^4 = r$ for all $r$ in $R$ must be commutative. But I don't see any non boolean example to ilustrate.
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1answer
47 views

Ideals of $\mathbb Z[x]$ containing $(3, x^3 - 1)$.

I would like to diagram the complete lattice of ideals of $R = \mathbb Z[x]$ containing the ideal $I = (3, x^3 - 1)$. By the lattice isomorphism theorem, each ideal of $R$ containing $I$ corresponds ...
2
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1answer
87 views

On finite subring of a division ring [closed]

If $R$ is a finite subring of a division ring $D$, then $R$ is a division ring or not ? Thanks.
2
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2answers
42 views

$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$

$$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$$ where $c \in \mathbb Z_9$, $w=e^{2\pi i/9}$ and $\mathbb Z_9$ is the ring of integers modulo 9.
2
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1answer
71 views

Structure of $(\mathbb{Z}/p^k\mathbb{Z})^\times$ other than via $p$-adics?

I think I heard somewhere long ago that $\mathbb{Z}/p^k\mathbb{Z}$'s unit group is cyclic if $p$ is odd and $C_2\times C_{2^{k-2}}$ if $p=2$. I remember trying to prove it and finding it surprisingly ...
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1answer
49 views

Is $\sum_{c \notin \mathbb Z_q}\psi(c)-\sum_{c \notin \mathbb Z_q}\psi(c^n)$ an integer?

Let $n$ be a positive integer. Let $p$ be an odd prime and $q=p^k$. Let $c \in \mathbb Z_q$. Consider the additive character $\psi:\mathbb Z_q \rightarrow \mathbb C^{\times}$ that is defined as ...
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2answers
68 views

Question concerning finite rings

Let $R$ be a finite ring. Is it possible that $R$ has an element $a\in R$ such that $a$ is a left divisor of zero and $a$ is not right divisor of zero? Thanks.
21
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1answer
697 views

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.
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1answer
42 views

On ring isomorphisms

Is it possible to have ring isomorphisms between some subsets of size $s^k$ of Galois ring $\Bbb Z_2^{s^k}$ and the full Galois ring $\Bbb Z_s^k$?
11
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1answer
220 views

Coprime elements in finite rings

Let $R$ be a finite commutative ring. Consider elements $a,b \in R$ such that $Ra+Rb=R$. A paper I'm reading asserts that there exists some $x,y \in R$ such that $x(a+yb) = 1$. Of course, it ...
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1answer
53 views

Relation between $a$ and $a^{-1}$ in integer rings about evenness

Could I ask something seemingly simple? Well, let $N$ be a positive odd number (the reason why I set $N$ to be odd is I could actually solve the problem when $N$ is even which is easy) and $a$ is an ...
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0answers
48 views

An extremal combinatorial problem over Finite rings

Let $q$ be an odd number and $g_i = (g_{i1} g_{i2} \dots g_{ir}) \in \Bbb Z_q^r$ a list of vectors with $i\in\{1,\ldots,L\}$. Let each $g_i$ have $0 < k < r$ zero entries. What is the maximum ...
2
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1answer
160 views

Are there any homomorphisms from integers into finite rings other than modulo $n$?

Are there any "homomorphisms" from $Z$ onto finite rings other than $Z/nZ$ ? I think if instead of mapping $k$ to $k$ (mod $p$), you map it to $p - (k$ (mod $p$)$)$ and you get $f(-ab) = f(a)f(b)$. ...
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2answers
109 views

Simple Combinatorics in finite rings

Let $g = [g_{1} g_{2} \dots g_{r}] \in \Bbb Z_{q}^{*r}$ be a given vector with each $g_{i} \in \Bbb Z_{q}^{*}$ where $\Bbb Z_{q}^{*}$ is $\Bbb Z_{q} \backslash \{0\}$ and $q > 6$ is odd. How many ...
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2answers
115 views

Presentation of finite rings (fields)

One knows that every finite group is isomorphic to a subgroup of $\operatorname{GL}(n)$ for some $n$ large enough. Can every finite ring be represented by a ring of matrices, i.e., is every ring ...