Chapter 5.1, Problem 52E

(See Exercise 51.)

a. Write out the elements of ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ and construct addition and multiplication tables for

this ring. (Suggestion: Write $0$ for $[0]$, $1$ for $[1]$ in ${\mathbb{Z}}_2$.)

b. Is ${\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$ a commutative ring?

c. Identify the unity elements, if one exists.

d. Find all units, if any exist.

e. Find all zero divisors, if any exist.

f. Find all idempotent elements, if any exist.

g. Find all nilpotent elements, if any exist.

Exercise 51.

Let $R$ and $S$

be arbitrary rings. In the Cartesian product $R\times S$ of $R$

and $S$, define

$(r,s)=(r^{\text{'}},s^{\text{'}})$ if and only if $r=r^{\text{'}}$ and $s=s^{\text{'}}$,

$(r_1,s_1)+(r_2,s_2)=(r_1+r_2,s_1+s_2)$,

$(r_1,s_1)\cdot (r_2,s_2)=(r_1{r}_2,s_1{s}_2)$.

Prove that the Cartesian product is a ring with respect to these operations. It is called the direct sum of $R$ and $S$ and is denoted by $R\oplus S$.

Prove that $R\oplus S$ is commutative if both $R$ and $S$ are commutative.

Prove $R\oplus S$

has a unity element if both $R$ and $S$

have unity elements.

Given as example of rings $R$ and $S$ such that $R\oplus S$ does not have a unity element.