I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's Algebra textbook (and loved it), but every book I have read afterwards on noncommutative algebra (Farb and Dennis' Noncommutative Algebra and T. Y. Lam's A First Course in Noncommutative Rings) have assumed that all rings are unitary. Could anyone give me a good reference please? Thank you all in advance!


2 Answers 2


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.

  1. Hungerford's Algebra
  2. Bresar's Introduction to Noncommutative Algebra.
  3. Behrens's Ring Theory.
  4. Warner's Modern Algebra (Section 32).
  5. Grove's Algebra (Chapter V).
  6. Herstein's Noncommutative Rings.
  7. McCoy's The Theory of Rings.
  8. Jacobson's Structure of Rings.
  9. Gardner and Wiegandt's Radical Theory of Rings.

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