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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Commutative ring of prime power order

Suppose $R$ is a non trivial commutative ring with identity of prime power order. What can we say about the structure of $R$? If $R$ is of prime order, then $R$ is a field?
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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 ...
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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, ...
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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 $n$?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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81 views

Any ring of prime order commutative ?

Is any ring of prime order commutative ?
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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 ...
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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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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 ...
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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
72 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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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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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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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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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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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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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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$R$ 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
50 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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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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$R$ 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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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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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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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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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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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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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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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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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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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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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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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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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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*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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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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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.