Chapter Questions
Give an example of a finite noncommutative ring. Give an example of an infinite noncommutative ring that does not have a unity.
The ring $\{0,2,4,6,8\}$ under addition and multiplication modulo 10 has a unity. Find it.
Give an example of a subset of a ring that is a subgroup under addition but not a subring.
Show, by example, that for fixed nonzero elements $a$ and $b$ in a ring, the equation $a x=b$ can have more than one solution. How does this compare with groups?
Prove Theorem $12.2$.
Find an integer $n$ that shows that the rings $Z_{n}$ need not have the following properties that the ring of integers has.a. $a^{2}=a$ implies $a=0$ or $a=1$.b. $a b=0$ implies $a=0$ or $b=0$.c. $a b=a c$ and $a \neq 0$ imply $b=c$. Is the $n$ you found prime?
Show that the three properties listed in Exercise 6 are valid for $Z_{p}$, where $p$ is prime.
Show that a ring is commutative if it has the property that $a b=c a$ implies $b=c$ when $a \neq 0$
Prove that the intersection of any collection of subrings of a ring $R$ is a subring of $R$.
Verify that Examples 8 through 13 in this chapter are as stated.
Prove rules 3 through 6 of Theorem $12.1$.
Describe all the subrings of the ring of integers.
Let $a$ and $b$ belong to a ring $R$ and let $m$ be an integer. Prove that $m \cdot(a b)=(m \cdot a) b=a(m \cdot b)$
Show that if $m$ and $n$ are integers and $a$ and $b$ are elements from a ring, then $(m \cdot a)(n \cdot b)=(m n) \cdot(a b)$. (This exercise is referred to in Chapters 13 and $15 .$ )
Show that if $n$ is an integer and $a$ is an element from a ring, then $n \cdot(-a)=-(n \cdot a)$