TRUE/FALSE Which of the following statements are valid for any ring R?
(a) For any $a, b, c, d \in R$, $(a + b)(c + d) = ac + ad + bc + bd$.
(b) For any $a, b\in R$, ${(a + b)}^2 = a^2 + 2ab + b^2$.
(c) If $a, b\in R$ satisfy $ab = 0$, then either $a = 0$ or $b = 0$.
(d) For any $a\in R$, $−(−a) = a$.
I believe that the first two are true because of the distributive property of rings. However, I am really not sure about the last two and can't think of any counter examples though I know their must exist some.
Any help would be much appreciated.
(a) is true (for the reason that you mentioned)
(b) is false. All you can say is that $(a+b)^2=a^2+ab+ba+b^2$. THis is different from $a^2+2ab+b^2$ if $a$ and $b$ don't commute.
(c) is false: take a ring with zero divisors;
(d) is true: since $a+(-a)=0$, by uniqueness of the opposite element you have $-(-a)=a$.