Operators on regular rings of Leavitt path algebras
Keywords:Leavitt path algebra, von Neumann regular ring, unit-regular ring, associated elements, partial order, shorted operator
In [8, Theorem 1], Jain and Prasad obtained a kind of symmetry of regular rings which is interesting and useful in the theory of shorted operators (cf. ). We show that this symmetry property indeed holds for endomorphism rings of Leavitt path algebras. Using this property, we analyze a (strong/weak) regular inverse of an element of the regular endomorphism ring A of the Leavitt path algebra L:= LK(E) (viewed as a right L-module). We also introduce some partial orders on the endomorphism ring A of the Leavitt path algebra L and investigate the behavior of regular elements in A.