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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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    $\begingroup$ Projective line :D $\endgroup$ – მამუკა ჯიბლაძე Jan 13 '18 at 17:40
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    $\begingroup$ @მამუკაჯიბლაძე how would you define $0\times\infty$? I understand from the question that the multiplication is everywhere defined. $\endgroup$ – YCor Jan 13 '18 at 17:57
  • $\begingroup$ @YCor You are right, I've not been attentive enough $\endgroup$ – მამუკა ჯიბლაძე Jan 13 '18 at 17:58
  • $\begingroup$ I take it you mean that when $x(y+z)$ is defined, then so is also $xy + xz$ and the two are equal. Right? $\endgroup$ – Todd Trimble Jan 13 '18 at 18:36
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    $\begingroup$ “Partial ring” is used in at least one place to mean something else, namely a ring with a partially defined multiplication which is commutative when defined. $\endgroup$ – Qiaochu Yuan Jan 13 '18 at 20:23

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