Name of $a*b=c$ and $b*a=-c$ $A_+=(A,+,0,-)$ is a noncommutative group where inverse elements are $-a$
$A_*=(A,*)$ is not associative and is not commutative
$\mathbf A=(A,+,*)$ is a structure where
1) if $a*b=c$ then $b*a=-c$ holds and
2) $(a*b)+a=b+(a*b)$

A-How is called the structure $\mathbf A$?
B-What is the name of 1) and 2) in abstract algebra (even not in the same structure)?
C-and what is this structure? It Has been already studied and does it have other intresting (or obvious) properties that I don't see?

Thanks in advance and I apologize for errors in my english.
Update
As Lord_Farin explained to me, $\mathbf A=(A,+,*)$ can't be a structure (closed) since the property 2) imply that $a*0$ and $0*a$ are assorbing elements of $(A,+)$ and that is impossible because $A_+$ is a non-trivial group (in the definition).
Anyways I notice that my questions B (about property 2) ) and C are still open in the case of a "generic" $A_+$ .

From 2) we have $(a*0)+a = 0+(a*0) = a*0 = (a*0)+0$ which contradicts the fact that $(A,+)$ is a group (which implies "left-multiplication" by $(a*0)$, i.e. $x \mapsto (a*0)+x$, is injective). Thus the structure $(A,+,*)$ cannot exist.
