Micheli Exercise 3.1.19.
Let $R$ be an integral domain, and $a\in R$ with $a\ne0.$
Show that if $ab=ac$ then $b=c.$
(5pts)
Guevara Proof
Micheli Exercise 3.1.20.
Let $R$ be a ring and $Z(R)$ the set of elements
$a\in R$ such that $ax=xa$ for any $x\in R.$
Show that $Z(R)$ is a subring of $R.$ This subring
is called the center of $R.$
(10pts)
Guevara Proof
Micheli Exercise 3.1.21.
Let $R$ be a finite integral domain, show that $R$ is a
field (hint: for any element $x$ consider the set of its
powers).
(10pts)
Guevara Proof
Micheli Exercise 3.1.22.
Let $R$ be a ring. Show that if $x\in R$ is nilpotent, then
$1+x$ is a unit and so is $1−x$
(hint: for example, notice that
$(x−1)(xn−1+xn−2+⋅ ⋅ ⋅+1) = \ldots).$
(10pts)
Alex Notes