if S is a ring with the property that s=s 2 for eachs element of S which of the

Question

if S is a ring with the property that s=s2 for eachs element of S which of the following must be true? I. s +s = 0 for each s element S II. (s + t)2 = s2 + t2 foreach s, t element of 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 Please explain Thanks if S is a ring with the property that s=s2 for eachs element of S which of the following must be true? I. s +s = 0 for each s element S II. (s + t)2 = s2 + t2 foreach s, t element of 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 Please explain Thanks

Explanation / Answer

x + x = (x +x)2 = x2 +2x2 + x2 =x + 2x + x = x +x + x + x

and since <S,+> is an abelian group, we cansubtract x + x from both sides of thisequation, which gives x + x = 0. A similarproof shows that every Boolean ring is commutative:

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