What is the name of $(\mathbb{Z}_2^s, \oplus, \odot)$ and where is it studied?

Question

I'm studying the ring $(\mathbb{Z}_2^s, \oplus, \odot)$, where $s$ is arbitrary, $\oplus$ is the sum modulo $2$, and $\odot$ is the AND.

Does it have a name? Even for a certain fixed $s>1$? Does anyone know of a book that studies its properties? Thanks

Answer 1

This is a special case of Boolean ring. Each element in this ring is a Boolean vector of length $s$.

Answer 2

It sure looks to me like you are talking about a finite Boolean ring. Every finite Boolean ring has this structure.

When the product is infinite, you still have a Boolean ring, but some Boolean rings are not just such a product. They are all subrings of such a product, though.