# Tagged Questions

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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### Help with understanding quotient ring structure

Let $R$ be the ring $\mathbb{Z}[x]/((x^2+x+1)(x^3+x+1))$ and $I$ be the ideal generated by $2$ in $R$. What is the cardinality of the ring $R/I$? I am having a hard time understanding what the ring ...
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### 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 $u_2$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 hint. ...
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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 and quotient rings of a finite 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 $I$ is an ideal of ...
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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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### 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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### 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 ...
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, $\alpha = \lfloor p/2 \rfloor$, ...
### 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 divisors ...