I've heard that in some book by someone named Kelley, perhaps an early edition of John L. Kelley's General Topology, the author gave a definition of a ring which turned out to be weaker than the usual definition. I seem to recall the problem involved taking some shortcut in stating the distributive law. I've heard that as a mean joke, people called this new mathematical structure a Kelley ring.
Is this true, and if so, can someone tell me what a Kelley ring is?
I dug up an old sci.math post from 9 June 1997 where I wrote:
A ring $R$ is a set equipped with operations $+$ and $\times$ and special elements $1$ and $0$, such that:
1) $+$ and $\times$ are both associative
2) $0$ is the unit for $+$ and $1$ is the unit for $\times$
3) every element $r$ of $R$ has an element $-r$ such that $r + -r = -r + r = 0$
4) $\times$ distributes over $+$ on both sides
Two amusing digressions for the experts... passed on to me by James Dolan. First, the usual definition of a ring also includes a clause saying that $+$ is commutative. However, this follows from the above definition. Second, there is a book by Kelley called General Topology, which is sort of the bible of basic topology. At some point Kelley had call to define a ring. In an early edition, he tried to save space by compressing clause 4) above. The usual version of 4) says that
$$r(t + u) = rt + ru$$
and
$$(r + s)t = rt + st$$
Kelley tried to say these both at once by saying that
$$(r + s)(t + u) = rt + ru + st + su $$
This obviously implies both of the above two by setting either $s = 0$ or $u = 0$. Right? Wrong! It would if we knew $0$ times anything is $0$. This follows from the usual definition of a ring, but showing that uses distributivity, and it does not follow from Kelley's version of distributivity. So Kelley's definition was not equivalent to the usual one. Just to annoy him, mathematicians wrote some papers on "Kelley rings", studying how his definition deviated from the usual one.
So, my old self answered my new self's question. I now think it's worth adding that in the 1963 reprinting of the 1955 edition of General Topology, and presumably any earlier edition, Kelley did not require his ring to have a multiplicative unit 1, so you can't use that.
However, by now I'm curious to see if this story is really true. Which edition of Kelley's book, if any, has this mistake? (Not the 1963 reprinting of the 1955 edition!) Or could the mistake be in his Introduction to Modern Algebra? Who, if anyone, wrote about "Kelley rings"? And what are they like?
This is an attempted answer to "what are Kelley rings like?"
To expand on Adam's comment in the same thread: in a Kelley ring (really a Kelley rng because having a multiplicative identity implies left and right distributivity in this context) x0 = (x+0)(0+0) = x0 + x0 + 00 + 00, so x0 = -(00 + 00). Therefore x0 = y0 (and 0x = 0y) for all x and y, and in particular x0 = 0x = 00. We also have 00 + 00 + 00 = 0. If 00 = 0, then we can prove distributivity: $x(y+z) = (x+0)(y+z) = xy + xz + 0y + 0z = xy + xz + 00 + 00 = xy + xz$. Therefore, in a Kelley rng which is not a rng, 00 is an element of order 3 in the additive group (which I haven't assumed to be commutative, by the way).
Adam's example ($\mathbb{Z}_3$ with $x \cdot y := 1$) can easily be shown to be the initial Kelley rng. You can make a Kelley rng from any group G with an element g of order 3 (or 1), setting $xy := g$. Call this $K(G, g)$. Then $K(G, g) \times K(H, h) \cong K(G \times H, (g, h))$. However, this doesn't hold for coproducts (e.g., $\mathbb{Z}_3$ with itself, ), so you could make more Kelley rngs that way.
I suppose there is also the free Kelley rng on a group. Beyond that, I can't really find any examples that aren't rngs. (The only rng contained in a non-rng Kelley rng is trivial: $0(0+0) = 0$ implies that $00 = 0$.) 0x not equaling 0 is one thing, but the condition that 0x has order exactly 3 just seems too weird to occur in nature.
By the way, the theory of ideals doesn't really work in a Kelley rng because the proof that multiplication of cosets is well-defined uses distributivity. You also need to assume that + is commutative for addition to be well-defined.
Edit: actually, multiplication of ideals is well-defined (assuming that + is commutative). But notice that a "kernel" is not $f^{-1}(0)$, it's $f^{-1}$ of the sub-Kelley ring containing 0. So an ideal in R should be a sub-k.r. closed under left and right multiplication in R, and also closed under addition of 00 and 00 + 00.
Then if $x-x', y-y' \in I$, $xy-(x'y') = xy + -(x'y) + x'y -(x'y')$. It's important to note here that $-(xy) \neq (-x)y$. They are, however, the same modulo addition of 00's. Then you use the "weak" distributive law $x(y+z) = xy + xz + 00 + 00$ and the proof goes through.
Edit2: Hmm, here: http://www.mathematik.uni-marburg.de/~gumm/Papers/Ideals%20in%20universal%20algebras.pdf is a paper which defines ideals for any universal algebra. Basically every set of the form $f^{-1}(0)$ (i.e. a congruence class of 0 for some congruence) is an ideal, but not necessarily vice versa. What I showed is that in a Kelley rng every normal subgroup containing 00 is a congruence class of 0. However, $\{0\}$ is a congruence class too, but it does not contain 00.
Note that under their definition of an ideal, an ideal I in a Kelley ring first of all has to be a normal subgroup. this ensures that addition of congruence classes is well-defined (you don't have to assume + commutative, clearly). Second, I satisfies $x_0y_0 + y_1x_1 + y_2x_2 \in I$ where the y's are in I and the x's are arbitrary (i.e. $xI + yI + zI \subseteq I$). Finally, we have $x_0y_0 - y_1x_1 \in I$, which given that I is normal implies that $xI = Iy$ for any x and y. As for multiplication: $(x + I)(y+I) = xy + xI + Iy + II \subseteq xy + RI + IR + IR \subseteq xy + I$, so $x-y \in I$ defines a congruence.
In the paper they give an equivalent condition for this: there exists a term $s(x,y)$ such that $s(x,x) = 0$ and $s(0,x) = x$. I guess this is just $y-x$, so that is actually sufficient.
1955 edition of General Topology
I wonder if the definition of field turns out to be correct.
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:
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 $R$.
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.
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 $b$.
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.
