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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The finite Ring 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 $Z(R)$ (Set of all zero divisors ...
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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
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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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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?
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77 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 ...
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1answer
29 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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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 ...
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1answer
45 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 $e$, ...
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1answer
67 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 $R\cong\mathbb{Z}_4$ ...
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90 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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44 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
37 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
52 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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40 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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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
74 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.
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1answer
66 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
90 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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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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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
44 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
38 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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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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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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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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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
45 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 ...
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1answer
78 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.
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2answers
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$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,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.
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1answer
67 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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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.
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682 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 $\Bbb Z_s^k$?
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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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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 ...
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1answer
151 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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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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106 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 ...
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Additive group of a finite ring of square free order is cyclic

$R$ is a finite ring of square free order $n>1$. How to show by the basis theorem for finite Abelian groups that additive group of $R$ is cyclic group of order $n$?
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Finite set of zero-divisors implies finite ring

Show that any commutative ring $R$ having only $n$ non-zero zero divisors ($n\geq 1$) is finite and doesn't contains more than $(n+1)^2$ elements.
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2answers
79 views

How to understand ideals in $F$, which is a finite commutative ring with $1$

I do not fully understand about ideals in finite rings, and I have to choose the correct answer to the following: If $F$ is a finite commutative ring with $1,$ then (i) Each prime ideal is a maximal ...
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1answer
106 views

Finite ring of sets

I have some questions about finite rings of sets and I'll be very grateful for any help. Let E be some fixed non-empty set. Suppose we are given some finite ring of subsets of set E, i.e. some ...
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345 views

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 ...
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1answer
454 views

Structure theorem of finite rings

Like structure theorem for finite abelian groups or modules over PID, is there any structure theorem for finite rings? Thanks.
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67 views

Sufficient condition for commutativity of finite rings

We know from Wedderburn's theorem any finite division ring is necessarily commutative. Is there any other condition on finite rings which forces the ring to be commutative?
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1answer
415 views

Problem on a finite commutative ring with no zero divisors [duplicate]

This is a problem from Dummit & Foote. Prove that a non-zero finite commutative ring that has no divisor is a field. (Do not assume the ring has a 1) Evidently, one has to use the theorem ...
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350 views

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 ...
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1answer
88 views

Definition of Bent functions over Galois rings using Fourier transform (walsh transform)

First, note that the following definition is true: Definition (Carlet): Let $R=GR(p^k,m)$. A function $f$ from $R^n$ to $R$ is bent if $$|\sum_{x \in R^n} w^{Tr(f(x)-ax)}|=|R|^{n/2}$$ where $a \in ...