László Fuchs’s contributions to commutative ring theory

In commemoration of the 90th birthday of Professor Laszlo Fuchs, this article gives a short account of some of his contributions to commutative ring theory.     

The socle series of a Leavitt path algebra

We investigate the ascending Loewy socle series of Leavitt path algebras L _K (E) for an arbitrary graph E and field K. We classify those graphs E for which L _K (E) = S _λ for some element S _λ of the Loewy socle series. We then show that for any ordinal λ there exists a graph E so that the Loewy length of L _K (E) is λ. Moreover, λ ≤ ω ₁ (the first uncountable ordinal) if E is a row-finite graph.     

Weakly Regular and Self-Injective Leavitt Path Algebras Over Arbitrary Graphs

We characterize the Leavitt path algebras over arbitrary graphs which are weakly regular rings as well as those which are self-injective. In order to reach our goals we extend and prove several results on projective, injective and flat modules over Leavitt path algebras and, more generally, over (not necessarily unital) rings with local units.     

Regularity Conditions for Arbitrary Leavitt Path Algebras

We show that if E is an arbitrary acyclic graph then the Leavitt path algebra L _K (E) is locally K-matricial; that is, L _K (E) is the direct union of subalgebras, each isomorphic to a finite direct sum of finite matrix rings over the field K. (Here an arbitrary graph means that neither cardinality conditions nor graph-theoretic conditions (e.g. row-finiteness) are imposed on E. These unrestrictive conditions are in contrast to the hypotheses used in much of the literature on this subject.) As a consequence we get our main result, in which we show that the following conditions are equivalent for an arbitrary graph E: (1) L _K (E) is von Neumann regular. (2) L _K (E) is π-regular. (3) E is acyclic. (4) L _K (E) is locally K-matricial. (5) L _K (E) is strongly π-regular. We conclude by showing how additional regularity conditions (unit regularity, strongly clean) can be appended to this list of equivalent conditions.     

Pure Submodules of Finite Direct Sums of Co-Purely Indecomposable Modules

A torsion-free module M of finite rank over a discrete valuation ring R with prime p is co-purely indecomposable if M is indecomposable and rank M = 1 + dim _R/pR (M/pM). Co-purely indecomposable modules are duals of pure finite rank submodules of the p-adic completion of R. Pure submodules of cpi-decomposable modules (finite direct sums of co-purely indecomposable modules) are characterized. Included are various examples and properties of these modules.     

*-Regular Leavitt path algebras of arbitrary graphs

If K is a field with involution and E an arbitrary graph, the involution from K naturally induces an involution of the Leavitt path algebra L _K (E). We show that the involution on L _K (E) is proper if the involution on K is positive-definite, even in the case when the graph E is not necessarily finite or row-finite. It has been shown that the Leavitt path algebra L _K (E) is regular if and only if E is acyclic. We give necessary and sufficient conditions for L _K (E) to be *-regular (i.e., regular with proper involution). This characterization of *-regularity of a Leavitt path algebra is given in terms of an algebraic property of K, not just a graph-theoretic property of E. This differs from the known characterizations of various other algebraic properties of a Leavitt path algebra in terms of graphtheoretic properties of E alone. As a corollary, we show that Handelman’s conjecture (stating that every *-regular ring is unit-regular) holds for Leavitt path algebras. Moreover, its generalized version for rings with local units also continues to hold for Leavitt path algebras over arbitrary graphs.     

The Dixmier-Moeglin Equivalence for Leavitt Path Algebras

Let K be a field, let E be a finite directed graph, and let L _K (E) be the Leavitt path algebra of E over K. We show that for a prime ideal P in L _K (E), the following are equivalent: