Leavitt path algebras: Graded direct-finiteness and graded Σ-injective simple modules

In this paper, we give a complete characterization of Leavitt path algebras which are graded Σ-V rings, that is, rings over which a direct sum of arbitrary copies of any graded simple module is graded injective. Specifically, we show that a Leavitt path algebra L over an arbitrary graph E is a graded Σ-V ring if and only if it is a subdirect product of matrix rings of arbitrary size but with finitely many non-zero entries over K or K[x,x−1] with appropriate matrix gradings. We also obtain a graphical characterization of such a graded Σ-V ring L. When the graph E is finite, we show that L is a graded Σ-V ring ⟺L is graded directly-finite ⟺L has bounded index of nilpotence ⟺ L is graded semi-simple. Examples show that the equivalence of these properties in the preceding statement no longer holds when the graph E is infinite. Following this, we also characterize Leavitt path algebras L which are non-graded Σ-V rings. Graded rings which are graded directly-finite are explored and it is shown that if a Leavitt path algebra L is a graded Σ-V ring, then L is always graded directly-finite. Examples show the subtle differences between graded and non-graded directly-finite rings. Leavitt path algebras which are graded directly-finite are shown to be directed unions of graded semisimple rings. Using this, we give an alternative proof of a theorem of Vaš [33] on directly-finite Leavitt path algebras.