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:
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.
- 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 $b$.
- 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 $S$.
- bof, an answer to the question Counterexamples in Algebra? on Math Overflow.