A semiprime ring $R$ is called a $\ast$-ring if the factor ring $R/I$ is in the prime radical for every nonzero ideal $I$ of $R$. A long-standing open question posed by Gardner asks whether the prime radical coincides with the upper radical $U(\ast _{k})$ generated by the essential cover of the class of all $\ast$-rings. This question is related to many other open questions in radical theory which makes studying properties of $U(\ast _{k})$ worthwhile. We show that $U(\ast _{k})$ is an N-radical and that it coincides with the prime radical if and only if it is complemented in the lattice $\mathbb{L}_{N}$ of all N-radicals. Along the way, we show how to establish left hereditariness and left strongness of important upper radicals and give a complete description of all the complemented elements in $\mathbb{L}_{N}$.

The main purpose of this paper is to give a new, elementary proof of Flanigan’s theorem, which says that a given ring A has a maximal essential extension ME(A) if and only if the two-sided annihilator of A is zero. Moreover, we discuss the problem of description of ME(A) for a given right ideal A of a ring with an identity.

Let ρ be a supernilpotent radical. Let ρ* be the class of all rings A such that either A is a simple ring in ρ or the factor ring A/I is in ρ for every nonzero ideal I of A and every minimal ideal M of A is in ρ. Let be the lower radical determined by ρ* and let ρφ denote the upper radical determined by the class of all subdirectly irreducible rings with ρ-semisimple hearts. Le Roux and Heyman proved that is a supernilpotent radical with and they asked whether if ρ is replaced by β, ℒ , 𝒩 or 𝒥 , where β, ℒ , 𝒩 and 𝒥 denote the Baer, the Levitzki, the Koethe and the Jacobson radical, respectively. In the present paper we will give a negative answer to this question by showing that if ρ is a supernilpotent radical whose semisimple class contains a nonzero nonsimple * -ring without minimal ideals, then is a nonspecial radical and consequently . We recall that a prime ring A is a * -ring if A/I is in β for every .

For any class of rings it is shown that the class (M) of all rings each nonzero homomorphic image of which contains either a nonzero -ideal or an essential ideal is a radical class. If is a class of simple rings the upper radical generated by , (M), is shown to be equal to (M) where ' is the class of simple rings complementary to .