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Abstract  

A non-simple idempotent *-ring with zero centre is constructed and a negative answer to a question of G. Tzintzis concerning a hypoidempotent radical property is given.

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Abstract  

A radical α in the universal class of associative rings is called matric-extensible if α (R n) = (α (R))n for any ring R, and natural number n, where R n denotes the nxn matrix ring with entries from R. We investigate matric-extensibility of the lower radical determined by a simple ring S. This enables us to find necessary and sufficient conditions for the lower radical determined by S to be an atom in the lattice of hereditary matric-extensible radicals. We also show that this lattice has atoms which are not of this form. We then describe all atoms of the lattice, and show that it is atomic.

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Summary  

We continue our study of the lattice of matric-extensible radicals of associative rings. We  find some atoms generated by simple rings of the lattices of all matric-extensible radicals, matric-extensible supernilpotent radicals and matric-extensible special radicals. We consider *-rings, which were previously defined by the second author, and consider when they generate atoms of these lattices.

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Abstract  

The base radical class L b(X), generated by a class X was introduced in [12]. It consists of those rings whose nonzero homomorphic images have nonzero accessible subrings in X. When X is homomorphically closed, L b(X) is the lower radical class defined by X, but otherwise X may not be contained in L b(X). We prove that for a hereditary radical class L with semisimple class S(R), L b(S(R)) is the class of strongly R-semisimple rings if and only if R is supernilpotent or subidempotent. A number of further examples of radical classes of the form L b(X) are discussed.

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