Rings and fields

Developing the idea of Example 2 from $\S 1$, show that the set $\mathcal{P}(\Omega )$ with the operations

is a ring with identity, all elements of whose additive group have order 2.

Establish the commutativity of an arbitrary ring in which every element $x$ satisfies the equation $x^{2}=x$. Is this true under the condition $x^{3}=x$?

Are the fields $\mathbb {Q}(\sqrt{2}), \mathbb {Q}(\sqrt{5})$ isomorphic?

Show that the epimorphic image of a commutative ring is a commutative ring.

Show that any finite integral domain $K$ is a field.

Let $p$ be a prime number and $K$ a commutative ring with identity such that $p x=0$ for all $x \in K$. Show that then

Prove that a ring $K$ consisting of five elements is either isomorphic to $\mathbb {Z}_{5}$, or is a ring with zero multiplication.

An element $x \neq 0$ of a ring $K$ is called nilpotent if $x^{n}=0$ for some $n \in \mathbb {N}$. Show that:

1. nilpotency of the element $x$ implies invertibility of the element $1-x$ in any ring with identity;

2. the ring $\mathbb {Z}_{m}=\mathbb {Z} / m \mathbb {Z}$ contains nilpotent elements precisely when $m$ is divisible by the square of a natural number $>1$.

Prove that in a ring $K$ with identity and of infinite cardinality $\left|K\right|$ there cannot be a finite number $n \geqslant 1$ of nonzero non-invertible elements.

Let $K$ be an arbitrary associative ring with identity 1, and let $a, b$ be its elements. Show that

where $d=1+b c a$, i.e., invertibility of $1-a b$ in $K$ implies invertibility of $1-b a$. What is the element $1+a d b$ equal to?

Show that the matrices $\left\Vert \begin{smallmatrix} a & b \\ -b & a\end{smallmatrix}\right\Vert$, where $b \in \mathbb {Z}_{3}$, form a field of nine elements, and that the multiplicative group of this field is cyclic of order 8.

Is the code $S_{0}$ (from Example 2 at the end of the section) able to correct two errors?