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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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The aim of this paper is to give a solution to the problem of the representability of quadratic functionals, acting on modules over *-rings, by symmetric (weak) sesquilinear forms. The obtained results generalize the one given by P. Šemrl and B. Zalar to a larger class of *-rings.

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Abstract  

Connections between radicals of alternative and right alternative rings are investigated, with emphasis on those which are nondegenerate in the sense that semi-simple rings have no absolute zero-divisors. In particular it is shown that nondegenerate radicals of right alternative rings have the Anderson-Divinsky-Sulinski property.

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] Călugăreanu , G. and Lam , T. Y. , Fine rings: A new class of simple rings , J. Algebra Appl. , 15 ( 9 ), ( 2016 ), 18 pages. [6] Diesl

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simple rings , Bull. London Math. Soc. , 42 no. 2 ( 2010 ), 191 – 194 . [6] C hen , W. , Polynomial rings over NLI rings need not be NLI , Studia Sci. Math

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