Endomorphism rings of Leavitt path algebras.
Let \(L=L_K(E)\) be the Leavitt path algebra of the graph \(E\) and \(A=\text{End}(L_L)\) the endomorphism ring of \(L\). One of the first results of the paper states that \(L\) is a cyclic left \(A\)-module if and only if it is a cyclic right \(A\)-module, if and only if the graph has a finite number of vertices. Since \(L\) can be embedded in \(A\), it has sense to investigate how much room does \(L\) occupy within \(A\). The authors prove that \(A\) is an algebra of right quotients of \(L\). Furthermore the Jacobson radical of \(A\) is zero and \(A\) is non-singular. The paper contains also the result that \(A\) is an exchange ring if and only if \(L\) is an exchange ring if and only if \(E\) satisfies condition (K). Also \(L\) is a projective left \(A\)-module and a pure left submodule of \(A\). Section 3 starts investigating some regularity conditions which pass from \(A\) to \(L\) (\(\pi\)-regularity, von Neumann regularity, weak regularity). Then, assuming row-finiteness of the graph \(E\), it is proved the equivalence of the following: \(A\) is von Neumann regular. \(E\) is acyclic and every infinite path ends in a sink. \(L\) is both left and right self-injective and von Neumann regular. \(L\) is a semisimple right \(L\)-module. -- As a corollary \(\text{End}(L_L)\) is von Neumann regular if and only if \(\text{End}(_LL)\) is von Neumann regular. It is known that a module \(M\) is quasi-injective if and only if \(M\) is invariant under every endomorphism of its injective hull. A generalization of this condition leads to the concept of an automorphism invariant module. A right module \(M\) over a ring \(R\) is said to be automorphism invariant if \(M\) is invariant under every automorphism of its injective hull \(E(M)\). It is proved in the paper that a right module \(M\) over a ring \(R\) is both continuous and automorphism invariant if and only if \(M\) is quasi-injective. Then, for a row-finite graph \(E\) they have the equivalence of: \(L\) is automorphism invariant as a right \(L\)-module. \(L\) is right continuous. \(L\) is right and left self-injective and von Neumann regular. \(E\) is acyclic and every infinite path ends in a sink. Also for a row-finite graph, it is proved in the paper the equivalence of: \(A\) is strongly \(m\)-regular for some positive \(m\). \(A\) is strongly \(\pi\)-regular. \(L\) is a direct sum (not necessarily finite) of matrix rings \(M_{n_i}(K)\) with \(n_i\leq m\) for a fixed integer (and for all \(i\)). \(E\) is acyclic, has no right infinite paths and no left infinite paths and there is a fixed positive integer \(m\) such that the number of paths ending in a given sink in \(E\) is \(\leq m\). Finally we mention the result that gives the equivalence of: \(A\) is left self-injective. \(L\) is semisimple and every homogeneous component is an Artinian ring. \(E\) is both row-finite and column finite, acyclic, and there are no left or right infinite paths in \(E\).
scientific article; zbMATH DE number 6476968
scientific article