Nucleus

Remarks

The sets where only the first, second or third condition is required are the left, middle and right nucleus. $N(A)$ is an $R$-submodule closed under the product, and the product on $N(A)$ is associative. $N(A) = A$ if and only if $A$ is associative. Example: if $A$ has a unit $1$, then $1 \in N(A)$, since $(1x)y = xy = 1(xy)$, and similarly in the other positions. Non-example: in the octonions, the basis element $i$ is not in the nucleus, because $(ij)\ell \neq i(j\ell)$ for the standard units $j$ and $\ell$. In fact the nucleus of the octonions is $\R 1$.