Lattices of radicals

Snider, Robert L.

doi:10.2140/pjm.1972.40.207

Cited by 57 publications

(44 citation statements)

References 12 publications

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“…It is well known [13] that the family K of all supernilpotent radicals is a lattice with respect to inclusion. Minimal elements of K are called supernilpotent atoms.…”

Section: Resultsmentioning

confidence: 99%

On Bad Supernilpotent Radicals

France-Jackson¹

2011

Bull. Aust. Math. Soc.

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A supernilpotent radical α is called bad if the class π(α) of all prime and α-semisimple rings consists of the one-element ring 0 only. We construct infinitely many bad supernilpotent radicals which form a generalization of Ryabukhin's example of a supernilpotent nonspecial radical. We show that the family of all bad supernilpotent radicals is a sublattice of the lattice of all supernilpotent radicals and give examples of supernilpotent radicals that are not bad.2010 Mathematics subject classification: primary 16N80.

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“…It is well known [13] that the family K of all supernilpotent radicals is a lattice with respect to inclusion. Minimal elements of K are called supernilpotent atoms.…”

Section: Resultsmentioning

confidence: 99%

On Bad Supernilpotent Radicals

France-Jackson¹

2011

Bull. Aust. Math. Soc.

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“…In [15] Snider proved that if ^, and ^2 are hereditary radicals, then (^, : !% 2 ) = {Fv | Stfi\R') c ^i(Fv') for every homomorphic image /?' of Fv} is the pseudocomplement of ^2 relative to ^, in the lattice of all radicals, that is, (£fr\ :<^?…”

Section: Proof We Shall Prove First That For Every Non-zero Ring R mentioning

confidence: 99%

“…terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700001695 [15] The singular ideal and radicals 209…”

Section: R(z) \ : E Z a (R) \ P(r)} Since A G Fi(r) There Exists X mentioning

confidence: 99%

The singular ideal and radicals

Ferrero

Puczyłowski

1998

J Aust Math Soc A

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“…The lattice of all radicals will be denoted by L and the lattice of left strong radicals by Ls. Recall ([9]) that a nonzero radical P is an atom of L (Ls) if and only if for every 0 = R ∈ P, P = l R (P = ls R ), where l R (ls R ) denotes the lower (lower left strong) radical determined by R. If R is a nonzero idempotent ring, then A ∈ l R (A ∈ ls R ) if and only if every nonzero homomorphic image of A contains a nonzero ideal (left ideal) which is a homomorphic image of R. Basic facts about radicals can be found in [2] and about their lattices in [10].…”

Section: Examplementioning

confidence: 99%