I have some questions about finite rings of sets and I’ll be very grateful for any help. Let E be some fixed non-empty set. Suppose we are given some finite ring of subsets of set E, i.e. some non-empty system $S \subset E$ such that $\forall A,B \in 2^E ~~ A \vartriangle B \in S$ […]
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 easy to develop for integral domains since these can be naturally embedded in their own fraction field. But what about rings having […]
Fields of finite order are well classified, and classification of groups of finite order has taken some depth in research. Why classification of finite rings and modules is not well studied in research?
Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime. My attempt: Let $I$ be an ideal of $R$. Then we have $I$ is maximal $\Leftrightarrow$ $R/I$ is a finite field $\Leftrightarrow$ $R/I$ is a finite integral domain $\Leftrightarrow$ $I$ is a prime ideal. Is […]
Show that any commutative ring $R$ having only $n$ non-zero zero divisors ($n\geq 1$) is finite and doesn’t contains more than $(n+1)^2$ elements.
Let $A$ be a ring with $0 \neq 1 $, which has $2^n-1$ invertible elements and less non-invertible elements. Prove that $A$ is a field.
Does a finite commutative ring necessarily have a unity? I ask because of the following theorem given in my lecture notes: Theorem. In a finite commutative ring every non-zero-divisor is a unit. If it had said “finite commutative ring with unity…” there would be no question to ask, I understand that part. What I’m asking […]
Let $R$ be a finite commutative ring. For $n>1$ consider the full matrix ring $M_n(R)$. For a matrix $A\in M_n(R)$ is true that the cardinality of the left annihilator (in $M_n(R)$) of $A$ equals the cardinality of the right annhilator?
I was struggling for days with this nice problem: Let $A$ be a finite commutative ring such that every element of $A$ can be written as product of two elements of $A$. Show that $A$ has a multiplicative unit element. I need a hint for this problem, thank you very much.
Like structure theorem for finite abelian groups or modules over PID, is there any structure theorem for finite rings? Thanks.