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.
221 questions
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Lifting central units modulo Jacobson radical
Let $R$ be a finite unital ring (not commutative) and $J$ it's Jacobson radical. Assume that $R$ is local, that is, $R/J$ is a division ring and hence a finite field of characteristic $p$.
Since $Q = (...
- 3,030
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1
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89
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Is the Beta index of the zero divisor graph of a finite commutative unital ring uniformly far from 1?
Is there a uniform upper bound $M<1$ for the Beta index of the zero divisor graph of an arbitrary finite unital commutative ring?
In the other word is there a positive number $...
- 851
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0
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42
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Uniqueness of common $ \mathbb Z _ n $-roots of polynomials in $ \mathbb Z [ x ] $ from linearity of their GCD
$ \def \Z {\mathbb Z} $EDIT: In the statement of the problem, the "every $ n \in \Z _ + $" is now changed to "all but finitely many $ n \in \Z _ + $". The unnecessary previous ...
- 6,896
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1
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50
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Every ideal of a finite principal ring is generated by a zero divisor
I stumbled upon this while doing introductory exercises to abstract algebra, but since i am still rather inexperienced in the subject, i would appreciate a second opinion. We will assume the ring to ...
- 713
1
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1
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178
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Linear congruences over a finite ring are equivalent
Let $n,N$ be two natural numbers. Let $a=(a_1, ..., a_n),b=(b_1, ..., b_n)\in (\mathbb{Z}/N\mathbb{Z})^n$ with $\gcd(a_1, ..., a_n, N) =1$ and $\gcd(b_1, ..., b_n, N) =1$. Define $H_a = \{(x_1,...,x_n)...
user1154495
2
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0
answers
124
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Subrings and quotients of finite semigroup algebras
Let $F = \Bbb F_p$ be a prime finite field and $R$ an arbitrary finite-dimensional associative (+ let's say unital) algebra over $F$. Then $R$ is a subalgebra (=subring here) of a matrix algebra $M_n(...
- 1,596
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5
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Why does Gaussian elimination sometimes work in rings where it should not?
I think it's best to illustrate this with an example.
Take for instance the ring of integers modulo $6$. If I have the system of equations:
$$ \begin{aligned} 2x + 2y &= 4 \\ 3x + 4y &= 3 \end{...
- 141
4
votes
1
answer
149
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A polynomial with unique root in every $ \mathbb Z _ n $
Let $ p ( x ) \in \mathbb N [ x ] $ be a polynomial with nonnegative integer coefficients, and $ a \in \mathbb Z $ be a given integer constant. If for all positive integers $ n $, $ p ( x ) + a $ has ...
- 6,896
2
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1
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695
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Classification of quadratic forms over $\mathbb{Z}/n\mathbb{Z}$ - even characteristic case
Let $R$ be a ring (unital, commutative) and $M$ a free $R$-module of finite rank. A quadratic form is a map $q:M\rightarrow R$ such that
$\forall r\in R:\forall m\in M: q(rm)=r^2\cdot q(m)$ and
the ...
- 2,024
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3
answers
2k
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Show that no ring of order 6 is an integral domain.
Problem Statement:
Show that no ring of order $6$ is an integral domain.
Some Definitions:
Integral Domain: a commutative ring with identity and no zero divisors.
$\mathbb{Z}_n$: group of the elements ...
- 141
3
votes
1
answer
158
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Solveablity of Diophantine equation over "computer numbers"
Hilbert's tenth problem asks whether there is an algorithm to determine if a given solution set to a Diophantine equation is non-empty. There is no such algorithm.
In practice for many engineering ...
- 256
2
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2
answers
175
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Diagonalizable binary matrices over $\mathbb{Z}_4$
I'm trying to figure out the following question:
Are symmetric, binary $n\times n$ matrices with zeros on the diagonal, are diagonalizable over $\mathbb{Z}_4$?
I know that it isn't true that $\textit{...
- 121
2
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0
answers
85
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Finite ring with irreducible element that is not prime
Let $R$ be a commutative unital ring. Let's call a non-zero non-unit $a \in R$
irreducible if $a=bc$ implies that (either) $b$ or $c$ is a unit,
prime if $a \mid bc$ implies that $a \mid b$ or $a \mid ...
- 349
5
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1
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503
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Structure Theorem for non-abelian finite groups or rings
How many structure theorems do we have in Abstract Algebra for finite algebraic structures? I know some of the following theorems:
If $G$ is a finite abelian group, then $G$ is a product cyclic ...
- 1,696
4
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1
answer
315
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On the definition of cohomological dimension
Let $G$ be a group and $R$ a commutative unital ring. We define the $R$-cohomological dimension of $G$ to be
$$cd_R(G) := \sup \{ n : H^n(G, M) \neq 0 \text{ for some } R[G]\text{-module } M \}.$$
I ...
- 3,085
2
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1
answer
240
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Let $(R, +, \cdot)$ be a finite ring without zero divisors, show that $R$ has a neutral element for $\cdot$. [duplicate]
I have to prove the question in the title, but I am having some difficulties.
Here's a sketch what I've already tried:
Choose $a \in R$.
Because $R$ is finite, there exist positive integers $i$ and $j$...
3
votes
1
answer
137
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Finite quotients of ring of integers of local field
Let $K$ be a non-Archimedean local field, so either a finite extension of $\mathbb{Q}_p$ or a finite extension of $\mathbb{F}_q((t))$. Let $\mathcal{O}$ denote its ring of integers and $\pi$ a ...
- 3,085
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If $x \in R$ is non-invertible implies $x^2 \in \{\pm x\}$ and $|R| >9$ odd then $R$ is a field
Let $(R, +, \cdot)$ be a commutative ring with $2n+1$ elements, for some $n\neq 4$ a positive integer. Suppose also that $R$ also satisfies the following condition: If an element $x\in R$ is non-...
- 257
3
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1
answer
143
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Polynomials for which the induced polynomial map is zero
Let $R$ be a commutative ring with $1$.
Out of curiosity, I wonder what is the state of art about $I_R=\{P\in R[X]\mid P(r)=0 \mbox{ for all }r\in R\}$.
This an ideal of $R[X]$, which can be rewritten ...
- 15.7k
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0
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40
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All roots of a polynomial in ring $F_2+uF_2+u^2F_2$, where $u^3=0$
Let $R=F_2+uF_2+u^2F_2$, where $u^3=0$, be a finite commutative ring. So $R=\{0,1,u,v,uv,u^2,v^2,v^3\}$, where $v=1+u$, $v^2=1+u^2$, $v^3=1+u+u^2$, $uv=u+u^2$. It is well known that
$$x^7-1=(x+v^3)(x^...
- 1
2
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3
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457
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$x^2+3x+3$ is irreducible in $\mathbb{F}_{25}[x]$
Give an example of an irreducible non-linear polynomial in $\mathbb{F}_{25}[x]$.
I know that $x^2+3x+3$ is irreducible in $\mathbb{F}_{25}[x]$ but I know no shorter proof then the exhaustive search (...
- 1,333
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2
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20
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Is it possible rewrite $\frac{\mathbb{Z}_m [x]}{<f(x)>}$ as a in direct sum for $m$ composite?
I have a doubt: Given $\frac{\mathbb{Z}_m [x]}{<f(x)>}$, where $m$ is composite and $f(x)=\prod_{i=1}^t f_i^{a_i} (x)$ (irreducible factors), can I admit $\frac{\mathbb{Z}_m [x]}{<f(x)>} \...
1
vote
1
answer
420
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Is a ring with order $2$ unique? (In the sense of isomorphism)
Apologies. My definition of a ring need not include the multiplicative identity.
I had this problem asking if there exists rings $R_1, R_2$ such that they both have order $2$ but they are not ...
- 3,946
3
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1
answer
60
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Prime ideals of $F_q[C_m]$
My main question is that what the prime ideals of group ring $F_q[C_m]$ are where $F_q$ is a finite field with $q$ elements and $C_m$ is a cyclic group of order m. To do this, I was thinking that how ...
- 1,470
0
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1
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437
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Existence of $n$ such that $a^n=a$ for all $a$ in $Z_m$ [duplicate]
This is a question from Contemporary Abstract Algebra which asks:
Find an integer $n > 1$ such that $a^n = a$ for all $a$ in $Z_6$.
Show that no such $n$ exists for $Z_m$ when $m$ is divisible ...
- 1,249
1
vote
1
answer
612
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Ring homomorphism from matrix ring to smaller ring
Let $\mathbb{F}$ be some finite field, and let $R := M_n(\mathbb{F})$ be the set of $n$-by-$n$ matrices over $\mathbb{F}$. Then $R$ is finite. Does there exist some pair $(\varphi, S)$ such that $S$ ...
- 804
1
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1
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60
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Ultimately show $R\cong S$ implies $R[x]\cong S[x]$.
The question above is going to be used to ultimately show that $R\cong S$ implies $R[x]\cong S[x]$. I understand how these results imply that, but I am having trouble actually doing the proof parts ...
- 361
1
vote
1
answer
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Constructing the group SL$_2$ of $\mathbb{Z}[i]$ mod $p$ in GAP
I'm trying to construct, in GAP, the group $$\mbox{SL}_2(\mathbb{Z}[i]/p),$$ where $p$ is a prime in $\mathbb{Z}[i]$, the Gaussian integers. $p$ could be real or complex (for $p$ real would be enough)....
- 153
2
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0
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38
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Show finite ring has identity when for each $x$ there exists $y$ such that $xyx = x$. [duplicate]
I'm working through an old qualifying exam problem, and I'm stuck:
Let $R \neq (0)$ be a finite ring such that for any element $x \in R$ there is $y \in R$ with $xyx = x$. Show that $R$ contains an ...
- 5,146
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1
answer
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GAP — GeneratorsOfRing giving a list with repeated element
I put the following code in GAP:
R := Integers mod 8;
and I get the answer:
[ ZmodnZObj( 1, 8 ), ZmodnZObj( 1, 8 ) ]
Can ...
- 455
2
votes
1
answer
58
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Find finite rings $(R,+,\times)$ such that for every unit $r$, $r-1$ is a unit except $r=1$.
Let $(R,+,\times)$ be a finite ring. $R^\times$ denotes the set of all invertible elements, i.e., units in $(R,\times)$.
Find finite rings $(R,+,\times)$ such that for every unit $r\in R^\times\...
- 1,174
4
votes
1
answer
486
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Finite Non Commutative Rings of Cardinality n
For any given $n\in N$, Can we find a non-commutative ring of $n$ elements (with or without identity)?
If not, can we find some condition on $n$ such that a non-commutative ring of $n$ elements ...
- 1,425
3
votes
3
answers
161
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How many solutions of $x^{p+1} \equiv 1 \mod p^{2017}$
How many solutions does $x^{p+1} \equiv 1 \mod p^{2017}$ have in set $\left\{0,1,...,p^{2017}-1 \right\}$?
$p$ is prime > 2.
My observations
$1$ is one of solutions of given equation.
$p$ is prime ...
user677339
0
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2
answers
752
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Ideals and order of a polynomial ring
Consider the ideal $I=(X^3+\hat2X+\hat1)$ of polynomial ring $R=\mathbb Z_3[X]$.
Is $R/I$ an integral domain?
How many elements does $R/I$ have?
Find the inverse of $X^3+\hat1$ in $R/I$.
(1) $(X^3+\...
- 113
4
votes
2
answers
53
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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
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2
answers
436
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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 ...
- 51
3
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2
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399
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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 ...
- 1,238
1
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2
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73
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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 $\...
- 624
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2
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279
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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 ...
- 1,351
5
votes
2
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69
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 ...
- 1,351
0
votes
1
answer
61
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 ...
- 6,773
-1
votes
2
answers
434
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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 ...
- 11
1
vote
4
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2k
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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,[...
- 435
5
votes
0
answers
138
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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 $R/I$ ...
- 3,970
3
votes
0
answers
102
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 $\phi\big(f(x)\big)=f(i)$. ...
- 939
2
votes
3
answers
678
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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 ...
- 14.8k
2
votes
1
answer
44
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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 ...
- 3,302
2
votes
2
answers
82
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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 ...
- 1,487
4
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0
answers
131
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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 ...
- 558
0
votes
1
answer
46
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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.