Name for a Boolean ring without a unit element

Question

Is there a standard name for a Boolean ring without a unit?

I read that historically ring and Boolean ring used to refer to possibly non-unital objects:

The old terminology was to use "Boolean ring" to mean a "Boolean ring possibly without an identity" -- https://en.wikipedia.org/wiki/Boolean_ring

Back in the day, the term ‘ring’ meant a possibly nonunital ring; that is a semigroup, rather than a monoid, in Ab. This terminology applied also to Boolean rings, and it changed even more slowly. -- http://ncatlab.org/nlab/show/Boolean+ring

Now that the name ring commonly assumes a unit element, is there a shorter name for a "possibly non-unital Boolean ring"

Answer 1

Since "rng" is standard for "ring without assuming an identity", "Boolean rng" seems a reasonable name that's likely to be understood, even though it doesn't seem to have been used much in the literature.

Answer 2

When I perform a Google books search for "Boolean ring", a majority of the top two pages of hits explicitly do not assume identity. The same is still true when restricting to hits with publication dates in the past decade. So it seems like a standard usage of boolean ring fits your requirements quite well already.

Naturally you are going to find category theorists preferring the version with identity, but this is not really conclusive evidence of a coherent trend. If you had come across sources written by operator algebraists, you would not have been lead down this path.

I did run into the same convention here, though. It reserves the term "Boolean algebra" for Boolean rings with identity. This is sensible because it can be considered as an algebra over the field of two elements only when a (unital) embedding of the field into the ring can be achieved.