Clarity on Boolean Algebra and Rings

Question

I'm trying to wrap my head around Abstract Algebra, Boolean rings, and it's a little difficult.

So I understand the ring (I believe it's a ring) <ℤ ,x, +, -, 0, 1 > is normal integer arithmetic stuff. We can add and multiply in it and still be in ℤ (All integers), but you can't divide because you may end up outside of ℤ.

So then my question is:

with the algebra (I don't believe it's a ring) <{0,1}^w, |, &, ~, 0^w, 1^w> where {0,1}^w denotes all strings of 1's and 0's of w length, and a^w denotes a string consisting of w repetitions of symbol a. are the | and & functions considered multiplication and addition? And if so why do they behave differently in the boolean algebra? because 1010₂ | 0111₂ = 1111₂ which is like 10₁₀ | 7₁₀ = 15₁₀. but 10₁₀ x 7₁₀ does not equal 15₁₀. so why would | be considered multiplication? (again if it actually is considered multiplication).

EDIT: You know a couple years back I hated math, now look where I am lmao.