TY - JOUR
T1 - Leavitt path algebras : graded direct-finiteness and graded Σ-injective simple modules
AU - Hazrat, Roozbeh
AU - Rangaswamy, Kulumani M.
AU - Srivastava, Ashish K.
PY - 2018
Y1 - 2018
N2 - 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 Lover 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 Lis a graded Σ-V ring ⇐⇒Lis graded directly-finite ⇐⇒L has bounded index of nilpotence ⇐⇒Lis 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 Lis a graded Σ-V ring, then Lis 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.
AB - 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 Lover 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 Lis a graded Σ-V ring ⇐⇒Lis graded directly-finite ⇐⇒L has bounded index of nilpotence ⇐⇒Lis 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 Lis a graded Σ-V ring, then Lis 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.
KW - algebra
KW - graded rings
KW - injective modules (algebra)
KW - rings (algebra)
UR - http://handle.westernsydney.edu.au:8081/1959.7/uws:45787
U2 - 10.1016/j.jalgebra.2018.01.041
DO - 10.1016/j.jalgebra.2018.01.041
M3 - Article
SN - 1090-266X
SN - 0021-8693
VL - 503
SP - 299
EP - 328
JO - Journal of Algebra
JF - Journal of Algebra
ER -