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- W2964007235 abstract "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." @default.
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- W2964007235 date "2018-06-01" @default.
- W2964007235 modified "2023-09-29" @default.
- W2964007235 title "Leavitt path algebras: Graded direct-finiteness and graded Σ-injective simple modules" @default.
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- W2964007235 doi "https://doi.org/10.1016/j.jalgebra.2018.01.041" @default.
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