Let $R$ be a ring. (Not necessarily commutative).
For every pair $(a,b)$ where $a , b \in R$ and $b \neq 0$, we define $S_{ab}$ as the set of pairs $(m,n) \in R \times R$ such that $an=bm$.
Let $F$ be the set of all distinct sets $S_{ab}$.
Further, operations $+$ and $*$ are defined on elements of $F$ as:
$$S_{ab}+S_{mn} = S_{xy}\;\; , \;\;\; x= an + mb, \;\;\; y=bn$$
$$S_{ab}*S_{mn} = S_{xy}\;\; ,\;\;\;\; x = am,\;\;\;\; y = bn.$$ Which of the following conditions is necessary and sufficient to make $F$ a field?
a) $R$ is a commutative ring but not an integral domain
b) $R$ is an integral domain but not a commutative ring
c) $R$ is a commutative integral domain
d) None of the above
This question was posed by a friend of mine. I guess it's from some entrance exam.
I believe it would be a field if $R$ is a commutative integral domain with $S_{0a}$ as the additive identity and $S_{aa}$ as the unity. The commutativity is needed for the unity to be well-defined and the integral domain structure for the additive identity. But I am still not very sure. The sets and elements mixture is confusing me.