Definition 2.8.10.  Let R be a ring. We define
1R
by:
1[R,⊕,0,⊖,⊙,1] := 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)