Definition talk:Unital Algebra
My suggestion is to define a unital algebra as one whose module and multiplication are unitary.
Why? Because it is convenient to reserve the simplest name for the most often used concept.
I went through 20-some books on abstract algebra. Those that do consider non-unitary modules at some point, only work with fields when they come to algebras (and vector spaces are by definition untary). For those that do not consider non-unitary modules, there is no question what "unital algebra" should mean.
I also found that words like "pre-unital" or "pseudo-unital" are used for different things that are not related to the module structure. Basically, there is zero indication that algebras over non-unitary rings or algebras with non-unitary underlying module over unitary rings are studied. --barto (talk) 06:28, 24 October 2017 (EDT)
This goes back to when I was trying to plunder that fascinating article of Baez's early in my mathematic enthusiasm, always promising to get back to it.
It may be the case that in order to have an identity, the ring needs to be unitary itself. I don't know. Maybe Baez takes it for granted that a ring is unital. Some sources do.