Ring

[vvaann]

If S is a ring with the property that s = s^2 for each s in S, which of the following must be true?

I. s + s = 0 for each s in S.
II. (s + t)^2 = s^2 + t^2 for each s, t in S.
III. S is commutative.

A. III only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III

Answer key:

SPOILER: E

[Dragonfinity]

This is a tough one.

We know that s = s^2 for every s in S. If, s,t are elements of S, then
(s + t) = (s + t)^2 = (s + t)(s + t) = s^2 + t^2 + st + ts = s + t + st + ts
Subtracting on both sides, we get
0 = st + ts
which implies that st = -ts. Hence,
(s + t)^2 = s^2 + t^2 + st + ts = s^2 + t^2 - ts + ts = s^2 + t^2,
so we've shown #2.

Further,
s + s = (s + s)^2 = s^2 + s^2 +s^2 + s^2 = s + s + s + s,
so subtracting we get
s + s = 0,
and we've shown #1.

Finally, s + s = 0 implies that s = -s for any s in S. Recall that we showed that for any s,t in S that st = -ts; therefore, st = -ts = ts. We've shown #3.

I had a really hard time working this problem out when practicing for the exam. Variations of this problem are in the free practice exam given by ETS and in the Princeton Review book.

[vvaann]

Dragonfinity,

You've had great explanations to the last two questions! They're perfectly correct, showing that you have very good skills on math.

Hope I can learn more from you.

Br,
vvaann,