Is there a name for, or notable structure that uses, weird "distributive laws" such as $a\times(b+c)=b\times a+c\times a$? Consider the following "multiplication" over "addition":
$$
a \times (b + c)
$$
The distributive law in common notion is the left distributive law:
$$
a \times (b + c) = a \times b + a \times c
$$
But what if?:
$$
a \times (b + c) = b \times a + c \times a
$$
$$
a \times (b + c) = a \times b + c \times a
$$
$$
a \times (b + c) = b \times a + a \times c
$$
Are there any name for these? I guess they're "anti", "exo", and "endo" distributive, respectively.
Are there any notable algebraic structure with any of these laws?
 A: My comment was a bit wrong, and too brief, so I'm going to expand this into a partial answer.
Suppose you have a set $R$ with operations $+$ and $\times$ such that $(R, +)$ is a group, and $\times$ left-endodistributes over $+$. Let $0$ be the identity under $+$.
Fix $a, b \in R$. Then
$$a \times 0 = a \times (0 + 0) = 0 \times a + a \times 0 \implies 0 \times a = 0.$$
We also have
$$0 = 0 \times (a + 0) = a \times 0 + 0 \times 0 = a \times 0,$$
and
$$a \times b = a \times (b + 0) = b \times a + a \times 0 = b \times a.$$
Therefore, $\times$ is commutative, and therefore distributes over $+$.
The same can be done with left-exodistributivity.
For left-antidistributivity, consider first
$$0 \times 0 = 0 \times (0 + 0) = 0 \times 0 + 0 \times 0 \implies 0 \times 0 = 0.$$
Next, note that
$$0 \times a = 0 \times (a + 0) = a \times 0 + 0 \times 0 = a \times 0.$$
Then,
$$(0 + a) \times (0 + a) = (0 + a) \times 0 + (0 + a) \times a = 0 \times 0 + 0 \times a + a \times 0 + a \times a,$$
which, when combined with the above identity, simplifies to $0 \times a + 0 \times a = 0$. But then,
$$0 = 0 \times a + 0 \times a = a \times (0 + 0) = a \times 0 = 0 \times a.$$
Finally, this again gives us that $\times$ is commutative and distributes over $+$, as
$$a \times (b + 0) = b \times a + 0 \times a = b \times a.$$
While this doesn't mean that left anti/endo/exo-distributivity properties are of no interest, it does mean that, in order to avoid "trivial" examples (i.e. ones where $\times$ distributes), we have to sacrifice a fair amount of structure of the additive magma, which means the result is not going to be as "ring-like" as you might have hoped.
