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, Amer. Math. Monthly 65 (1958), 289:

> In John L. Kelley, *General Topology*, p. 18, a definition of a ring is given in 
which the left and right distributive laws are replaced by the composite distributive 
law: $(u+v)(x+y)=ux+uy+vx+vy$.  
(a) Show by an example that such a system is not necessarily a ring.  
(b) Show that if such a system contains an element $a$ such that $a0=0$ (in 
particular, if the system has a multiplicative identity), then the system is a 
ring.

3. R. A. Beaumont, Postulates for a ring (Solution of Problem 4784), Amer. Math. Monthly 66 (1959), 318.

> **Summary by me:** For part (a) Beaumont takes an additive group of order $3$ and defines the product of every pair of elements to be the same nonzero element $u$. For part (b) he uses the "composite distributive law" to show that $a0=0$ implies $b0=0b=0$ for all $b$.

4. Toru Saito, Note on the distributive laws, Amer. Math. Monthly 66 (1959), 280-283.

> **Summary by me:** The author defines $w$-rings and $c$-rings. A $c$-*ring* is a "Kelley ring". A $w$-*ring* is a system $(S,+,\cdot)$ which is an abelian group with respect to addition, a semigroup with respect to multiplication, and contains fixed elements $e_1,e_2$ such that
$$x(y+z)=xy+xz-e_1,\quad(y+z)x=yx+zx-e_2,\quad\text{for all }x,y,z\in S.$$
He shows that, in a $w$-ring $S$, we have $e_1=00=e_2$; this element is called the *defining element* of the *w*-ring $S$. The order of the defining element with respect to the additive group of $S$ is called the the *order* of $S$. After proving some results on the existence and structure of $w$-rings, he relates $c$-rings (Kelley rings) to $w$-rings with the following:
> 
> THEOREM 3. A $c$-ring is a $w$-ring of order $3$ or $1$ according as $00\ne0$ or $00=0$ and $00$ is the defining element of the $w$-ring. Conversely, a $w$-ring of order $3$ or $1$ is a $c$-ring.

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