Direct sum of two rings. Let S1 and S2 be any two rings, and consider the set of all symbols (s1, s2) where s1 ε S1, s2 ε S2. If we define addition and multiplication of these symbols by

(s1, s2) + (t1, t2) = (s1 + t1, s2 + t2),

and

(s1, s2)(t1, t2) = (s1t1, s2t2),

it is easy to verify that this set becomes a ring S which we call the direct sum of S1 and S2, and denote by S1 + S2.

Clearly, the zero element of S = S1 + S2 is (0, 0), the first 0 being the zero of S1, the second the zero of S2. If S1 and S2 have unit elements e1 and e2 respectively, then S has the unit element (e1, e2). If both S1 and S2 have more than one element, S has proper divisors of zero since

(s1, 0) (0, s2) = (0, 0)

for every s1 in S1, s2 in S2. Furthermore, S is a commutative ring if and only if both S1 and S2 are commutative.