Let be an algebra, not necessarily associative multiplicatively. The nucleus of is:
where is the associator bracket. In other words, the nucleus is the set of elements that multiplicatively associate with all elements of . An element is nuclear if
.
Accompanying the concept of a nucleus is that of the center of a nonassociative algebra (which is slightly different from the definition of the center of an associative algebra):
![$\displaystyle \mathcal{N}(A):=\lbrace a\in A\mid [a,A,A]=[A,a,A]=[A,A,a]=0 \rbrace,$](http://images.planetmath.org:8080/cache/objects/6548/l2h/img3.png "$\displaystyle \mathcal{N}(A):=\lbrace a\in A\mid [a,A,A]=[A,a,A]=[A,A,a]=0 \rbrace,$"){kind=small}
![$ [\ , , ]$](http://images.planetmath.org:8080/cache/objects/6548/l2h/img4.png "$ [\ , , ]$"){kind=small}
![$\displaystyle [ab,c,d]$](http://images.planetmath.org:8080/cache/objects/6548/l2h/img12.png "$\displaystyle [ab,c,d]$"){kind=small}
![$ [c,ab,d]=[c,d,ab]=0$](http://images.planetmath.org:8080/cache/objects/6548/l2h/img19.png "$ [c,ab,d]=[c,d,ab]=0$"){kind=small}
![$\displaystyle \mathcal{Z}(A):=\lbrace a\in \mathcal{N}(A)\mid [a,A]=0 \rbrace,$](http://images.planetmath.org:8080/cache/objects/6548/l2h/img22.png "$\displaystyle \mathcal{Z}(A):=\lbrace a\in \mathcal{N}(A)\mid [a,A]=0 \rbrace,$"){kind=small}
![$ [\ , ]$](http://images.planetmath.org:8080/cache/objects/6548/l2h/img23.png "$ [\ , ]$"){kind=small}
