Vol. 40, No. 1, 1972

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Lattices of radicals

Robert L. Snider

Vol. 40 (1972), No. 1, 207–220

Nothing first that the class of all radicals for associative rings forms a lattice under a natural ordering, we show that several important subclasses, including the class of hereditary radicals, form sublattices. We give an example showing that the special radicals do not form a sublattice even though they form a complete lattice under the same ordering.

After we have studied various lattice-theoretic properties of our main lattices, showing, in particular, that the lattice of hereditary radicals is Brouwerian, we determine the atoms of that lattice and show that it has no dual atoms by computations with free rings. We characterize pseudocomplements in the lattice of hereditary radicals and give partial results toward determining which radicals of that lattice are complemented.

Mathematical Subject Classification
Primary: 16A21
Received: 5 November 1970
Revised: 19 August 1971
Published: 1 January 1972
Robert L. Snider