Definition 2.8.4.  Let R be a ring, a, b ∈ R. We define:
a + b := a ⊕ b if R = [R, ⊕, 0, ⊖, ⊙, 1] (R is a set, ⊕ : R × R → R is an operation on R, 0 ∈ R, ⊖ : R → R is a function, ⊙ : R × R → R is an operation on R, 1 ∈ R such that (R, ⊕, 0, ⊖, ⊙, 1) forms a ring)
Remarks.

This definition lets us add elements without decomposing the ring.