Definition 2.8.3.  Let R be a ring. We define
(R)
by:
([R, ⊕, 0, ⊖, ⊙, 1]) := R  (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)
We write “let a ∈ R” for “let a ∈ (R).”
Remarks.

This definition retrieves a representative carrier set from a ring (which is actually an equivalence class of rings), and enables typical abuse of notation.