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:

1. 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 $R$.

2. D. W. Jonah, [Problem 4784](http://www.jstor.org.www2.lib.ku.edu/stable/2310260 ), Amer. Math. Monthly 65 (1958), 289.

3. R. A. Beaumont, Postulates for a ring ([Solution of Problem 4784](http://www.jstor.org.www2.lib.ku.edu/stable/2309657 )), Amer. Math. Monthly 66 (1959), 318.

4. Toru Saito, [Note on the distributive laws](http://www.jstor.org.www2.lib.ku.edu/stable/2309634 ), Amer. Math. Monthly 66 (1959), 280-283.

5. bof, [an answer](http://mathoverflow.net/questions/29006/counterexamples-in-algebra/166233#166233) to the question [Counterexamples in Algebra?](http://mathoverflow.net/questions/29006/counterexamples-in-algebra) on Math Overflow.