The answer is yes (unless I made a mistake somewhere).

For example, you can replace addition with the operation $a\oplus b=a+b-1$. This is a commutative, associative binary operation with identity $1$ and the inverse of $a$ is given by $2-a$.

You replace multiplication by $a\odot b= a+b-ab$. This is a commutative, associative binary operation with identity $0$.

All that remains is to show that the associative laws hold.

The answer is yes (unless I made a mistake somewhere).

For example, you can replace addition with the operation $a\oplus b=a+b-1$. This is a commutative, associative binary operation with identity $1$ and the inverse of $a$ is given by $2-a$.

You replace multiplication by $a\odot b= a+b-ab$. This is a commutative, associative binary operation with identity $0$.

All that remains is to show that the distributive laws hold.