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)