Problem $1$: The following statement is either true or false. If the statement holds in general, write True and prove it $($you can refer the proof in the textbook$).$ Otherwise, state an example to disprove it$.$

A$.Z_{19}$ is a field.

B$.$ In a ring $(a+b{)}^2=a^2+2ab+b^2$ for $a,b$ elements of a ring.

C$.R[x]$ is a Principal Ideal Domain.

D$.6$ is a zero divisor of $Z_{15}.$

E$.$ The only ideal of $Q$ is $Q.$

F$.$ If an ideal I of $R$ contains a unit, then $I=R.$