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Is there a name for finite-dimensional associative $F$-algebras having the Jacobson radical of codimension 1. Of course they are particular local algebras and, indeed, the converse is true provided $F$ is algebraically closed.

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Those are local basic algebras.

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  • $\begingroup$ A finite dimensional algebra is basic if modulo its radical is a direct product of copies of the base field, and in that case it is local iff there is exactly one factor. $\endgroup$ – Mariano Suárez-Álvarez Apr 20 '13 at 16:12
  • $\begingroup$ Some people would say split basic and use basic for the radical quotient being a direct product of division rings. $\endgroup$ – Benjamin Steinberg Apr 20 '13 at 18:07

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