This paper owes its origins to Pere Menal and his work on Von Neumann Regular (= VNR) rings, especially his work listed in the bibliography on when the tensor product K = A ⊗K B of two algebras over a field k are right self-injective (= SI) or VNR. Pere showed that then A and B both enjoy the same property, SI or VNR, and furthermore that either A and B are algebraic algebras over k (see [M]). This is connected with a lemma in the proof of the Hilbert Nullstellensatz, namely a finite ring extension K = k[a1, ..., an] is a field only if a1, ..., an are algebraic over k.
@article{urn:eudml:doc:41734,
title = {Self-injective Von Neumann regular subrings and a theorem of Pere Menal.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {541-567},
mrnumber = {MR1209823},
zbl = {0782.16003},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41734}
}
Faith, Carl. Self-injective Von Neumann regular subrings and a theorem of Pere Menal.. Publicacions Matemàtiques, Tome 36 (1992) pp. 541-567. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41734/