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.
4
votes
2answers
45 views
Stuck: Finding an Isomorphism for an Invertible Ring
I'm stuck on a problem creating an isomorphism between rings. Specifically, let $\mathbb{Z}[\sqrt{7}] = R$.
Then for the invertible group $(R/3R)^\times$, I want to find an isomorphism to another ...
4
votes
2answers
232 views
Do there exist finite commutative rings with identity that are not Bézout rings?
A similar question has been asked before: Example of finite ring which is not a Bézout ring, but has not been answered.
There also seems to be a dearth of resources online regarding this ...
2
votes
1answer
47 views
If $R$ is a finite ring, then $\exists_{n>m>0}: x^n=x^m$ for all $x\in R$
I need some help for the following proof:
If $R$ is a finite ring, then $\exists_{n>m>0}: x^n=x^m$ for all $x\in R$.
I feel there's one or more little tricks to use to see how you get to ...
-1
votes
1answer
47 views
A finite ring which contains a field [closed]
Prove or disprove: If $A$ is a finite ring such that there exists a field $K,$ $K \subset A,$ then $|A|$ is a power of $|K|.$
1
vote
2answers
38 views
Finding inverses in quotient rings
In $ A=\mathbb{Z}[i]=\{a+bi \ : \ a,b \in \mathbb{Z}\} $ we consider $a=7+56i; \ b=3+3i; \ c=1+8i$. We will write $(a)$ to refer to the ideal generated by $a$
Find out whether the elements $\...
-1
votes
2answers
95 views
How to show that $2$ is invertible in a ring with odd cardinality?
Let $R$ be a commutative ring with unity that has an odd number of elements. Show that $2$ is invertible in $R$.
Attempt
I've found that $2 \ne 0$ from Lagrange, since the order of $1$ in the ...
5
votes
2answers
57 views
The number of polynomial functions $f:A\to A$ is $|A|^2$ if and only if $x^2=x$ for all $x\in A$.
Let $A$ be a commutative ring with $n$ elements, $n\ge2$. Prove that the next statements are equivalent:
$(\forall x\in A)(x^2=x)$.
The number of polynomial functions $f:A\to A$ is $n^2$.
I managed ...
0
votes
1answer
48 views
Is $\frac{R[x]}{(f(x))}$ finite for $R$ finite and $f$ not monic polynomial?
If $R$ is a finite commutative ring, then is $\frac{R[x]}{(f(x))}$ finite with $f$ not monic polynomial?
I can prove above claim if f(x) is monic polynomial using division algorithm? But I am not ...
-1
votes
2answers
26 views
How to solve systems of linear equations over a finite ring [closed]
I don't know where to start and how to go forth when solving system of equations in for example $\mathbb{Z}_{11}$. I have 2 different systems I want help with with a walkthrough to understand what is ...
1
vote
4answers
141 views
How to prove $\Bbb Z[i]/(1+2i)\cong \Bbb Z_5$? [duplicate]
How to prove $\Bbb Z[i]/(1+2i)\cong \Bbb Z_5$?
My method is
$$(1+2i)=\big\{a+bi丨a+2b≡0\pmod 5\big\},$$
So any $a+bi$ in $\Bbb Z(i)$,we got
$$a+bi=(b-2a)i+a(1+2i).$$
So $\Bbb Z[i]/(1+2i)=\big\{0,[...
5
votes
0answers
74 views
Let $R$ be a finitely generated subring of a number field. Is $R/I$ finite for every non-zero ideal of $R$?
Given any finitely generated subring $R$ of a number field (finite extension of $\mathbb{Q}$) or a global function field (finite extension of $\mathbb{F}_p(T)$), does $R$ have the property that $R/I$ ...
3
votes
0answers
71 views
Show that $\Bbb Z_p[i]$ is isomorphic to $\Bbb Z_p[x]/\langle x^2+1\rangle$.
Let $p$ be a prime number. Show that $\Bbb Z_p[i]$ is isomorphic to $\Bbb Z_p[x]/\langle x^2+1\rangle$.
My attempt:
Define
$$\phi : \Bbb Z_p[x] \to \Bbb Z_p[i]$$
by $\phi\big(f(x)\big)=f(i)$. ...
2
votes
3answers
115 views
Factor $x^ 5 - x^4 - x^ 2 - 1$ modulo $16$, and over $\mathbb{Q }$
From ARTIN algebra books chapter $12$ question $4.19$:
Factor $x^ 5 - x^4 - x^ 2 - 1$ modulo $16$, and over $\mathbb{Q }$
My works : I have check in $\mathbb{Z}_{16}$ as $x^ 5 - x^4 - x^ 2 - 1$ is ...
2
votes
1answer
19 views
if a ring is finite then the translation $x\rightarrow ax$ is surjective where $a\in A$ is regular
In a proof of the inversibility of regular elements in a finite ring, there is the following argument:
let $A$ be a finite ring and $a\in A$ regular .
the translation $A\rightarrow A: x\rightarrow ...
2
votes
2answers
71 views
What is the name of $(\mathbb{Z}_2^s, \oplus, \odot)$ and where is it studied?
I'm studying the ring $(\mathbb{Z}_2^s, \oplus, \odot)$, where $s$ is arbitrary, $\oplus$ is the sum modulo $2$, and $\odot$ is the AND.
Does it have a name? Even for a certain fixed $s>1$? Does ...
4
votes
0answers
105 views
Subring of $\text{Mat}_n(Z_m)$ is commutative if $x^2=0 \implies x=0$.
Let $A$ be subring of $\text{Mat}_n(Z_m)$. Suppose, for $x\in A$, $x^2=0$ implies $x=0$.
Claim A is commutative.
Attempt
$A$ is finite ring, hence Artinian.
If it is possible to claim that Jacobson ...
0
votes
1answer
36 views
How do I find a ring with a primary ideal having n elements?
I would like to know how can I find a ring with (at least) a primary ideal which has n elements (not generators, but elements) for a given n ?
Thank you.
-1
votes
1answer
34 views
Minimal Polynomials over an extension field [closed]
Is there any easy way to solve this?
Find the minimal polynomial of $(\sqrt{2})$+$(\sqrt[5]{3})$ over $\mathbb{Q}$.
5
votes
1answer
86 views
Galois Theory for Finite Local Commutative Rings
Let $R\subseteq S$ be two finite commutative local rings with unique maximal ideals $m$ and $M$, respectively. We say that $S$ is a separable extension of $R$ if $mS=M$. We also say that $S$ is a ...
5
votes
2answers
222 views
Minimal ideal in commutative finite rings
Let $R$ be a commutative finite ring with identity, and let $I$ be a minimal ideal of $R$, that is, a non-zero ideal that there is no ideal strictly between $I$ and $0$. Now let $\{I_i\}_{i\in A}$ be ...
2
votes
1answer
82 views
Identifying the ring $R=\mathbb{Z}_{9}[x]/(x^2-3,3x)$
I want to know if there is a simple form of the ring
$$R=\mathbb{Z}_{9}[x]/(x^2-3,3x)$$
I tried to start with the equations $3x\equiv0$ and $x^2\equiv 3$. So, $3x^2\equiv 0$ and $3x^2\equiv 9$. ...
5
votes
1answer
75 views
Classify all finite rings such that each unit has order 24
Problem: Suppose $R$ is a finite (associative) ring with 1 such that every unit of $R$ has order dividing 24. Classify all such $R$.
My attempt: I had to quotient out the jacobson radical $J(R)$ so ...
0
votes
1answer
35 views
solutions to $u^2 + u = 0$ in quotient ring $GF(2)[x]/p(x)$
In a finite field $GF(2^n)$, $u^2+u=0$ has only two solutions: $u=0, u=1$. (Not sure I can prove why this is true, probably something to do with invertibility.)
In a quotient ring $GF(2)[x]/p(x)$ ...
0
votes
1answer
119 views
Question related to Ideals of the ring of all functions from the set $\{1,2, \dots , 10\}$ to $\mathbb Z_2 $. [duplicate]
I have the following question in a competitive exam , but I failed to answer it.The question is:
Let $\mathcal R = \{f:\{1,2, \dots , 10\} \rightarrow \mathbb Z_2\} $ be the set of all $\mathbb Z_2$...
2
votes
2answers
127 views
On polynomials over finite ring
Let $R$ be a finite commutative ring with unity ; then does there exist a non-empty proper subset $A \subseteq R$ and $f(X) \in R[X]$ such that $f(r)=1 , \forall r \in A$ and $f(r)=0 , \forall r \...
2
votes
1answer
45 views
Enumerating finite local commutative rings effectively
Background
I am interested in writing a program to extend the sequence Number of commutative rings with 1 containing n elements. This sequence is not as well linked-to as the other three of its type, ...
2
votes
1answer
46 views
Extension of isomorphism of submodules
If $V$ is an $n$-dimensional vector space over $\mathbb{F}_q$, and $V_1$ and $V_2$ are subspaces of $V$ (both k-dimensional), there exists an isomorphism $L: V \to V$, where if $u \in V_1$, then $L(u) ...
5
votes
1answer
102 views
Zero divisors on finite rings
I'm trying to prove or disprove the following fact:
Let $R$ be a finite ring and $x \in R$ be a zero divisor. If $x$ is not nil-potent, then for some $n > 1$ we have $x^n = x$, i.e., $x$ is $n$-...
0
votes
1answer
82 views
Why is $\mathbb{Z}_m$ a commutative ring?
I understand for it to be a ring it has to be closed under the addition and product, have inverses for the additive group, distributive and identity in the product.
I can see that it is closed ...
0
votes
3answers
355 views
For a finite local ring $R$, why $|R|$ is a prime power. [closed]
When I read about finite local rings, every one firstly suppose that $|R|=p^t$ where $R$ is a finite local ring and $p$ is a prime. I don't understand why this can be supposed ?
0
votes
1answer
2k views
Finite integral domain is a field: proof condition
Theorem: a finite integral domain is a field
proof:
Let D be a finite integral domain with unity 1. Let a be any non-zero element of D.
If a=1, a is its own inverse and the proof concludes.
Suppose,...
0
votes
1answer
86 views
Find finite rings $(R, +, \times$) such that $(R,+)$ is cyclic.
I'm finding finite rings $(R, +, \times$) such that $(R,+)$ is cyclic. $\mathbb{Z}_n$ is a good example.
Up to ring isomorphism, is there any exmaples other than $\mathbb{Z}_n$ ?
Thanks.
For ...
0
votes
0answers
42 views
What is the difference between $Z/(4)$ and the field $F4$? [duplicate]
I assumed that if we quotient $Z$ by the ideal generated by 4, it will be the same as the field $F4$. It turns out not to be the case because $Z/(4)$ is not a field.
In general, when is $Z/(n) = Fn$?
1
vote
1answer
1k views
Non isomorphic rings of order 4
How do I show that $\mathbb{Z}_4$, $\mathbb{Z}_2 \times \mathbb{Z}_2$, $L=\left\{\left(\begin{smallmatrix} a & b \\ 0 & a \end{smallmatrix}\right)\mid a,b\in\mathbb{Z}_2\right\}$ and $\mathbb{...
1
vote
0answers
34 views
Examples of Finite Non-Unital Integral Rings [duplicate]
Here, integral rings are rings without nonzero zero-divisors. In this question, rings are not assumed to be unital (i.e., they may not have the multiplicative identity). Some people call them rngs.
...
3
votes
1answer
263 views
How to draw a sublattice to exhibit diagonalization?
Given the matrix:
$$
A=\begin{pmatrix}
3 & 1 \\
-1 & 2 \\
\end{pmatrix}.
$$
Let $V =\mathbb Z^2$ and $L = AV$. We want to find basis for $V$ and $L$ and ...
0
votes
1answer
252 views
How to describe a ring after adjoining an element to it?
I want to know how to go about describing a ring after adjoining an element that satisfies a certain relation.
As an example, I'm considering the ring obtained from Z3 by adjoining an element a ...
0
votes
3answers
210 views
How to find the GCD of $29-3i$ and $11+10i$
I'm trying to find the GCD of $29-3i$ and $11+10i$ in $Z[i]$
I know that one way I can do this, is to factorize each of the numbers and find the GCD through that. But the norms in these two numbers ...
1
vote
0answers
297 views
How to describe all ideals in a quotient ring?
$\newcommand{\C}{\mathbb{C}}$I'm a bit confused by quotient rings, and I want to know the general technique for describing their ideals.
For example, say I have the ring $\C[x]/(x^3)$
If we consider ...
0
votes
2answers
472 views
Definition of a Quotient Ring (need clarification).
My understanding is that we can view the definition of a Quotient ring $R/I$ as a set of cosets.
For example, the ring $Z/(6)$ which I believe is $Z6$, can be viewed like this:
$(6) + 0 = \{...,-12,...
0
votes
2answers
106 views
Factoring polynomials in $\mathbb Q[x]$ and $\mathbb Z[i]$
I'm practicing reducing polynomials in different rings, but I'm stuck on a few. I would appreciate any help.
1) I want to factor $7+i$ in $\mathbb Z[i]$. The norm is $50$, so we know it's reducible....
1
vote
0answers
104 views
Prove that $\mathbb{Z}[i]$ is a euclidean domain (intuition behind the geometric proof)
I am viewing the proof that $\mathbb{Z}[i]$ is a Euclidean domain, but I'm having a very hard time imagining it geometrically. Or in other words, showing that the size function exists by viewing the ...
1
vote
1answer
110 views
why is the set of intersections and symmetric differences of a fixed set finite
I have a question very similar to Finite ring of sets
For a given set U, the set R of subsets is a ring under the operations of symmetric difference ($\bigtriangleup$) and intersection ($\cap$).
...
1
vote
1answer
405 views
How many isomorphism classes of associative rings (with identity) are there with 35 elements?
How many isomorphism classes of associative rings (with identity) are there with 35 elements?
The underlying additive group $G$ is of the form $pq$ ($p=7, q=5$). So $G$ is either of form $\Bbb Z/35\...
7
votes
2answers
496 views
Is a p-Sylow subgroup of the underlying additive group of a finite commutative ring an ideal?
Let $R$ be a finite commutative ring. Let $H$ be a $p$-Sylow subgroup of the additive group $(R,+)$. Is $H$ an ideal in $R$?
For ex:
R= Z/6Z is a finite commutative ring.
H={0,3} is a 2-sylow ...
1
vote
0answers
30 views
Any hint with the next theorem?
"Let $GR=(s,sm)$" the Galois Ring with characteristic $p^s$ and $p^{sm}$ elements. If $n\mid m$ then exists $R_0=GR(s,sn)$ a Galois Ring that contains $R$ as a subring".
I've already seen that this ...
0
votes
1answer
65 views
Why $\mathbb{Z}_p$ over $\mathbb{Z}_{p^2}$ cannot be free?
Makoto Kato in the answer to this question wrote:
Let $p$ be a prime number.
Let $R = \mathbb{Z}/p^2\mathbb{Z}$.
Let $M = R/pR$.
Since the number of elements of $M$ is $p$, $M$ cannot be free....
0
votes
1answer
56 views
A relation in a finite ring [closed]
Let $R$ be a finite ring such that for any $a,b\in R$ there exists $c\in R$ (depending on $a$ and $b$) such that $a^2+b^2=c^2$.
Prove that for any $a,b,c\in R$ there exists $d\in R$ such that $2abc=...
1
vote
1answer
64 views
$R$ be a commutative ring with unity , $|R^{\times}|=p$ an odd prime then is $R$ isomorphic to a quotient of $\mathbb Z_2[x]/\langle x^p-1\rangle$?
Let $R$ be a commutative ring with unity such that its group of invertible elements has order $p$ an odd prime , then is it true that there exist a surjective ring homomorphism from $\mathbb Z_2[x]/\...
1
vote
1answer
547 views
Proving every nonzero element in a finite ring is either a unit or a zero divisor
I've just started learning ring theory, and the book I'm using uses "Every nonzero element in a finite ring is either a unit or a zero divisor" implicitly without explanation. So I came across this ...
