Notes on Simple Modules over Leavitt Path Algebras

Given an arbitrary graph E and any field K, a new class of simple left modules over the Leavitt path algebra L of the graph E over K is constructed by using vertices that emit infinitely many edges. The corresponding annihilating primitive ideals are described and is used to show that these new class of simple L-modules are different from(that is non-isomorphic to) any of the previously known simple modules. Using a Boolean subring of idempotents induced by paths in E, bounds for the cardinality of the set of distinct isomorphism classes of simple L-modules are given. We also append other information about the Leavitt path algebra L(E) of a finite graph E over which every simple left module is finitely presented.Comment: 17 page

The Theory of Prime Ideals of Leavitt Path Algebras over Arbitrary Graphs

Given an arbitrary graph E and a field K, the prime ideals as well as the primitive ideals of the Leavitt path algebra L_K(E) are completely described in terms of their generators. The stratification of the prime spectrum of L_K(E) is indicated with information on its individual stratum. Necessary and sufficient conditions are given on the graph E under which every prime ideal of L_K(E) is primitive. Leavitt path algebras of Krull dimension zero are characterized and those with various prescribed Krull dimension are constructed. The minimal prime ideals of L_K(E) are are described in terms of the graphical properties of E and using this, complete descriptions of the height one as well as the co-height one prime ideals of L_K(E) are given.Comment: 32 page