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Depth of Idempotent‐Generated Subsemigroups of a Regular Ring
Author(s) -
Hannah John,
O'Meara K. C.
Publication year - 1989
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-59.3.464
Subject(s) - mathematics , idempotence , semigroup , ring (chemistry) , pure mathematics , field (mathematics) , algebra over a field , organic chemistry , chemistry
If S is an idempotent‐generated semigroup, its depth is the minimum number of idempotents needed to express a general element as a product of idempotents. Here we study the depth of S where S is the semigroup generated by all the idempotents of a von Neumann regular ring R , and the depth of various subsemigroups of S . For example, if R is directly finite, the depth of S equals the index of nilpotence of R , which considerably extends a result of Ballantine (1978) for matrices over a field. We also answer a query of Professor Howie by supplying a ring‐theoretic explanation of Reynolds' and Sullivan's (1985) result that the depth is 3 for certain subsemigroups in the infinite‐dimensional full linear case.