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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-6
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1answer
115 views

Prove $R$ is a finite ring [on hold]

Suppose that $R$ is a ring and $D(R)$ is the set of all elements $x\in R$ such that there exist $b,c\neq 0$ with $xb=cx=0$. Prove that if $1<|D(R)|<\infty$ then $|R|<\infty$. ($|S|<\infty$ ...
3
votes
3answers
79 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) ...
0
votes
0answers
84 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 ...
2
votes
1answer
56 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 ...
0
votes
1answer
39 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 $u_2$ ...
1
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0answers
101 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
34 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 ...
1
vote
1answer
42 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.
1
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1answer
35 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 $uz=vz$ ...
0
votes
1answer
23 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 $\{0\}$, ...
2
votes
0answers
39 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
votes
1answer
61 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 ...
1
vote
2answers
57 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 ...
3
votes
3answers
120 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
votes
1answer
90 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
votes
1answer
53 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
votes
1answer
81 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
votes
2answers
59 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
votes
0answers
31 views

On units in subrings ( or ideal ) and quotient ring of 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 $S$ is an ideal ...
0
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0answers
34 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 ...
0
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1answer
51 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 ...
0
votes
2answers
47 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 ...
2
votes
2answers
57 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
votes
2answers
175 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
votes
1answer
69 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 ...
1
vote
1answer
103 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
67 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 ...
4
votes
1answer
74 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
104 views

Any ring of prime order commutative ?

Is any ring of prime order commutative ?
0
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1answer
29 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
votes
0answers
87 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 $N$, ...
1
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0answers
76 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
votes
1answer
35 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 ...
1
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1answer
75 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, $\alpha = \lfloor p/2 \rfloor$, ...
0
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0answers
46 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 ...
1
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0answers
51 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 ...
0
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0answers
35 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 ...
1
vote
3answers
82 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?
2
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1answer
111 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 ...
1
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1answer
54 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 ...
2
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2answers
67 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
50 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
votes
1answer
101 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
votes
2answers
100 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 ...
1
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2answers
68 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 ...
1
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1answer
98 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 ...
1
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2answers
62 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 ...
1
vote
2answers
91 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 ...
0
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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 ...