Were there ever considered sets $P$ with associative multiplication and a partially defined commutative, associative addition, $+: U\to P,$ $U\subset P\times P$, such that $x(y+z)=xy+xz$ when the left side is defined? Let's call them partial rings.

I am studying a certain ring (the coordinate ring of a certain algebraic variety) which is hard to describe explicitly but which has the interesting property that it is a union of partial subrings $P$, each of which nicely mapping to a polynomial ring. I wonder what is a significance of that.

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