Found problems: 6
Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\
$\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\
$\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\
$\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true?
\\ \\
[i] (Bogdan Blaga)[/i]
Let $\alpha \in \mathbb{C} \setminus \mathbb{Q}$ be such that the set $A= \{ a+b \alpha : a,b \in \mathbb{Z} \}$ is a ring with respect to the usual operations of $\mathbb{C}.$ If the ring $A$ has exactly four invertible elements, prove that $A= \mathbb{Z}[i].$
A ring $A$ is called Boolean if $x^2 = x$ for every $x \in A$. Prove that:
a) A finite set $A$ with $n \geq 2$ elements can be equipped with the structure of a Boolean ring if and only if $n = 2^k$ for some positive integer $k$.
b) The set of nonnegative integers can be equipped with the structure of a Boolean ring.
Determine all non-trivial finite rings with am unit element in which the sum of all elements is invertible.
[i]Mihai Opincariu[/i]
Consider an odd prime $p$. A comutative ring $(A,+, \cdot)$ has the property that $ab=0$ implies $a^p=0$ or $b^p=0$. Moreover, $\underbrace{1+1+\cdots +1}_{p \textnormal{ times}} =0$. Take $x,y\in A$ such that there exist $m,n\geq 1$, $m\neq n$ with $x+y=x^my=x^ny$, and also $y$ is not invertible. \\ \\
$\textbf{(a)}$ Prove that $(a+b)^p=a^p+b^p$ and $(a+b)^{p^2}=a^{p^2}+b^{p^2}$ for all $a,b\in A$.\\
$\textbf{(b)}$ Prove that $x$ and $y$ are nilpotent.\\
$\textbf{(c)}$ If $y$ is invertible, does the conclusion that $x$ is nilpotent stand true?
\\ \\
[i] (Bogdan Blaga)[/i]
We say a ring $(A,+,\cdot)$ has property $(P)$ if :
\[
\begin{cases}
\text{the set } A \text{ has at least } 4 \text{ elements} \\
\text{the element } 1+1 \text{ is invertible}\\
x+x^4=x^2+x^3 \text{ holds for all } x \in A
\end{cases}
\]
a) Prove that if a ring $(A,+,\cdot)$ has property $(P)$, and $a,b \in A$ are distinct elements, such that $a$ and $a+b$ are units, then $1+ab$ is also a unit, but $b$ is not a unit.
b) Provide an example of a ring with property $(P)$.