As you can see from the other answers, a so-called "Kelley ring" is a ring (without identity) in which the usual distributive laws are replaced by the identity $(u+v)(x+y)=ux+uy+vx+vy$. Toru Saito calls them $c$-rings in the note listed below. Here is a short bibliography of this topic:
- John L. Kelley, General Topology, D. Van Nostrand Company, Inc., Princeton, New Jersey, 1955. The following quotation is from p. 18 of the July, 1957, reprinting:
A ring is a triple $(R,+,\cdot)$ such that $(R,+)$ is an abelian group and $\cdot$ is a function on $R\times R$ to $R$ such that: the operation is associative, and the distributive law $(u+v)\cdot(x+y)=u\cdot x+u\cdot y+v\cdot x+v\cdot y$ holds for all members $x$, $y$, $u$, and $v$ of $R$.
D. W. Jonah, Problem 4784, Amer. Math. Monthly 65 (1958), 289.
R. A. Beaumont, Postulates for a ring (Solution of Problem 4784), Amer. Math. Monthly 66 (1959), 318.
Toru Saito, Note on the distributive laws, Amer. Math. Monthly 66 (1959), 280-283.
bof, an answer to the question Counterexamples in Algebra? on Math Overflow.