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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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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. –  Mariano Suárez-Alvarez Apr 20 '13 at 16:12
    
Thank you very much! –  Sammy Apr 20 '13 at 17:36
    
Some people would say split basic and use basic for the radical quotient being a direct product of division rings. –  Benjamin Steinberg Apr 20 '13 at 18:07

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