Direct Sums of Rings
One way of constructing new rings from existing one is through the direct sum of rings. Recall that the cartesian product of two sets A x B consists of the collection of ordered pairs {(a,b):a ∈ A, b ∈ B}. If R and S are rings we can form a new ring out the cartesian product in the following manner. Let R x S the cartesian product of R and S considered just as sets. On the set R x S define
(r1,s1) + (r2,s2) = (r1 + r2) + (s1 + s2)
and
(r1,s1) ⋅ (r2,s2) = (r1 r2, s1s2)
Under the operations defined above R x S becomes a ring. The zero element is (0,0) where the first 0 is the zero element of R and the second 0 is he zero element of S. We call this ring direct sum of the rings R and S denoted by R + S. R + S is commutative if and only if both R and S are commutative.
We can extended the direct sum to any finite number of rings R1, ..., Rn.
We shall define an analogous notion, the direct product(sum) of groups.