On coatoms of the lattice of matric-extensible radicals

Abstract

A radical α in the universal class of all associative rings is called matric-extensible if for all natural numbers n and all rings A, A ∈ α if and only if Mn(A) ∈ α, where Mn(A) denotes the n × n matrix ring with entries from A. We show that there are no coatoms, that is, maximal elements in the lattice of all matric-extensible radicals of associative rings.