Books on Rings without Identity
Jacobson's Structure of rings develops a bit of ring theory without assuming identity. Also Gardner and Wiegandt's book Radical Theory of Rings does not assume identities.
Any book on $C^*$ algebras would also have to deal with rings missing identity.
- Hungerford's Algebra
- Bresar's Introduction to Noncommutative Algebra.
- Behrens's Ring Theory.
- Warner's Modern Algebra (Section 32).
- Grove's Algebra (Chapter V).
- Herstein's Noncommutative Rings.
- McCoy's The Theory of Rings.
- Jacobson's Structure of Rings.
- Gardner and Wiegandt's Radical Theory of Rings.