The Upper Radical Property and Lower Radical Property of Groups

We take in this paper an arbitrary class [Formula: see text] of groups as a base, and define a radical property 𝒫 for which every group in [Formula: see text] is 𝒫-semisimple. This is called the upper radical property determined by the class [Formula: see text]. At the same time, we define a radical property 𝒫 for which every group in [Formula: see text] is a 𝒫-radical group. This is called the first lower radical property determined by the class [Formula: see text]. Also, we give another construction leading to the second lower radical property which is proved to be identical with the first one.

ON SUPERNILPOTENT NONSPECIAL RADICALS

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 $\mathcal {L}\left ( \rho ^{\ast }\right ) $ 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 $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a supernilpotent radical with $\rho \subseteq \mathcal {L}\left ( \rho ^{\ast }\right ) \subseteq \rho _{\varphi }$ and they asked whether $\mathcal {L} \left ( \rho ^{\ast }\right ) =\rho _{\varphi }$ 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 $\mathcal {L}\left ( \rho ^{\ast }\right ) $ is a nonspecial radical and consequently $\mathcal {L}\left ( \rho ^{\ast }\right ) \neq \rho _{\varphi }$. We recall that a prime ring A is a * -ring if A/I is in β for every $0\neq I\vartriangleleft A$.

A generalisation of the lower radical class

In this work we demonstrate that the lower radical class construction on a homomorphically closed class of associative rings generates a radical class for any class of associative rings. We also give a new description of the upper radical class using the construction on an appropriate generating class.

Essential covers and complements of radicals

We show that a radical has a semisimple essential cover if and only if it is hereditary and has a complement in the lattice of hereditary radicals. In 1971 Snider gave a full description of supernilpotent radicals which have a complement. Recently Beidar, Fong, Ke, and Shum have determined radicals with semisimple essential covers. Using their results, we are able to provide a lower radical representation of complemented subidempotent radicals. This completes Snider's description of hereditary complemented radicals.

On a class of radicals of rings

Let λ be a property that a lattice of submodules of a module may possess and which is preserved under taking sublattices and isomorphic images of such lattices and is satisfied by the lattice of subgroups of the group of integer numbers. For a ring R the lower radical Λ generated by the class λ(R) of R-modules whose lattice of submodules possesses the property λ is considered. This radical determines the unique ideal Λ (R) of R, called the λ-radical of R. We show that Λ is a Hoehnke radical of rings. Although generally Λ is not a Kurosh-Amitsur radical, it has the ADS-property and the class of Λ-radical rings is closed under extensions. We prove that Λ (Mn (R)) ⊆ Mn (Λ(R)) and give some illustrative examples.

On prime essential rings

A ring A is prime essential if A is semiprime and every prime ideal of A has a nonzero intersection with each nonzero ideal of A. We prove that any radical (other than the Baer's lower radical) whose semisimple class contains all prime essential rings is not special. This yields non-speciality of certain known radicals and answers some open questions.