Weak ring which is not a ring I am reading N. Hindman and D. Strauss's book "Algebra in the Stone-Cech Compactification". In chapter 16 they define weak ring as:

If $(S,+,•)$ is a left weak ring then both $(S,+)$ and $(S,•)$ are semigroups (not necessarily either of them are commutative) and $•$ is left distributive over +. (Meaning $a•(b+c)=a•b+a•c$.) 

Also they say in that book that every ring is a weak ring, and I have understand all that, but they didn't give an example of a weak ring which is not a ring. I tried but couldn't find any. So please help me on this.
 A: One example would be the countably infinite ordinals, that is, the set of ordinals $\alpha$ with $\omega \le \alpha < \omega_1$.
Both the sum and the product of countably infinite ordinals is again countably infinite. Both addition and multiplication of ordinals are associative, so we indeed have semigroups. And ordinal multiplication is left-distributive over ordinal addition. Thus the countably infinite ordinals indeed form a left weak ring.
And indeed, that's all they form:
Neither $0$ nor $1$ are infinite, so we indeed have just semigroups, not even monoids. And for infinite ordinals, neither addition nor multiplication is commutative. We don't even have right-distributivity.
Edit:
In case you want both left and right associativity (that is, the structure being both a left and a right weak ring), take the set of $n\times n$ real matrices ($n>1$) with all entries being strictly positive. Again, it is not hard to show that this set is closed under addition and multiplication, and both matrix addition and matrix multiplication are associative, thus making addition and multiplication a semigroup. And of course, matrix multiplication distributes over matrix addition on both sides.
Again, neither the zero matrix, nor the identity matrix is in that set, because both have entries that are zero (the identity matrix is why I added the condition $n>1$; if $n=1$, the identity matrix obviously has $1$ as the only element), so neither addition nor multiplication is a monoid. And matrix multiplication is not commutative, and that is also true if we restrict it to matrices with strictly positive entries.
A: A slight modification of the authors' suggestion can produce non-commutative examples, namely to take subsets of a non-commutative ring that happen to be closed under addition and multiplication.  For example a proper two-sided ideal of a non-commutative ring.
For the sake of concreteness, consider the ring of two-by-two integer matrices, a non-commutative ring.  From this ring take the subset $S$ of matrices whose entries are even.  Clearly $S$ is closed under addition and multiplication, and the properties of left- and right-distributivity are inherited from the larger ring.
It lacks a multiplicative identity of course, so it isn't a proper ring.
