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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!

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up vote 5 down vote accepted

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.

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  1. McCoy's The Theory of Rings.
  2. Herstein's Noncommutative Rings.
  3. Warner's Modern Algebra.
  4. Bresar's Introduction to Noncommutative Algebra.
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