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.
1 Answer
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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. $\endgroup$ Apr 20, 2013 at 16:12
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$\begingroup$ Some people would say split basic and use basic for the radical quotient being a direct product of division rings. $\endgroup$ Apr 20, 2013 at 18:07