On Monday, February 4, 2013 10:55:03 AM UTC-6, Arturo Magidin wrote:
> The only two books I have that do not allow 1=0 are: Zariski and Samuel's "Commutative Algebra", which restricts the use of the term "identity" to rings that are not nullrings; and Lam's "A First Course in Noncommutative rings" and "Lectures on Rings and Modules", which specifies this explicitly in the introduction.
Added: Note, however, that the zero ring does not qualify as a "ring" under Lam's definitions, so it cannot be an example. So the only book I have that both allows the 0 ring, and *does not* recognize the 0 ring as a ring with identity, is Zariski-Samuel. It also happens to be by far the oldest, having been published originally in 1958.