An interesting problem about a "quasi ideal"

Question

Let R be an infinite ring without an unity. Let I ⊆ R satisfying: for any a,b ∈ I, r ∈ R, we have a+b ∈ I, ar ∈ I, ra ∈ I. Then there are two possibilities about I:
(1) I must be an ideal, i.e., for any a ∈ I, we have -a ∈ I;
(2) there exists b ∈ I such that -b ∉ I.
Which one is right?

Thanks in advance.