Definition 2.8.14.1.  Let R be a ring, S ⊆ (R). We define:
S is a subring of R  :⇔  [∀ a, b ∈ S : a + b ∈ S] and 0R ∈ S and [∀ c ∈ S : c ∈ S] and [∀ d, e ∈ S : d ⋅ e ∈ S] and 1R ∈ S
References.