This investigation is motivated by a hoped-for application to the study of the possible forms of the monoid of isomorphism classes of finitely generated projective modules over a von Neumann regular ring; but that goal remains distant.
We end with a normal form result for the algebra obtained by tying together a k-algebra R given with a nonzero element p satisfying 1∉pR+Rp and a k-algebra S given with a nonzero q satisfying d70e8d862c5f568a8a4e6" title="Click to view the MathML source">1∉qS+Sq, via the pair of relations p=pqp, q=qpq.