Let $a \oplus b := \gcd(a,b)$ for $a,b \in \mathbb{N_0}$. Starting from the observation that for each $a,b,c \in \mathbb{N_0}$ we have:
$$a \gcd(b,c) = \gcd(ab,ac)$$
we might translate this to the distributive law:
$$a \cdot (b \oplus c) = (a\cdot b) \oplus (a\cdot c)$$
The set $\mathbb{N_0}$ with $\oplus$ forms a commutative monoid and we might construct its Grothendieck-Group. Setting $$[(a,b)]\times [(c,d)] := [(ac \oplus bd, ad \oplus bc)]$$ and $$[(a,b)] \oplus [(c,d)] := [(a \oplus c, b \oplus d)]$$
my question is, if this makes the Grothendieck-Group to a ring? If so, what properties does this ring have? Can we further localize the non-zero elements of this ring to make it to a field?
Thanks for your help!