A new setting for constructing von Neumann regular rings

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ABSTRACTA classical and clever construction by George Bergman in the mid 1970s was of a unit-regular algebra Q (over an arbitrary field F) that contains a regular subalgebra R which is not unit-regular. The algebra Q was constructed inside a full linear ring of an uncountable-dimensional vector space over F. Here we give a much simpler construction of Bergman’s R and Q inside an algebra B of countably-infinite matrices. The algebra B does not seem to have been noticed before, possibly because it is not obviously a ring. It also suggests possibilities for answering the difficult open problem of constructing a non-separative regular ring.