We verify that the $k$-nil radical of an obstinate 
(resp. associative, strong, weakly implicative,
implicative, sub-implicative, sub-commutative) ideal is an
obstinate (resp. associative, strong, weakly implicative,
implicative, sub-implicative, sub-commutative) ideal. 
We prove that every $k$-nil radical of a $q$-ideal and 
an $a$-ideal is also a $q$-ideal and an $a$-ideal.