Operators on regular rings of Leavitt path algebras

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. [9]). 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.