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Lia Vas

Publications and source records attributed to Lia Vas.

34 records · Page 2Linked to original sources
arXiv2013

Noetherian Leavitt path algebras and their regular algebras

In the past, it has been shown that the Leavitt path algebra $L(E)=L_K(E)$ of a graph $E$ over a field $K$ is left and right noetherian if and only if the graph $E$ is finite and no cycle of $E$ has an exit. If $Q(E)=Q_K(E)$ denotes the regular algebra over $L(E),$ we prove that these conditions are further equivalent with any of the following: $L(E)$ contains no infinite set of orthogonal idempotents, $L(E)$ has finite uniform dimension, $L(E)$ is directly finite, $Q(E)$ is directly finite, $Q(E)$ is unit-regular, and a few more equivalences. In addition, if the involution on $K$ is positive definite, these conditions are equivalent with the following: the involution $^\ast$ extends from $L(E)$ to $Q(E),$ $Q(E)$ is $^\ast$-regular, $Q(E)$ is finite, $Q(E)$ is the maximal (total or classical) symmetric ring of quotients of $L(E),$ every finitely generated nonsingular $L(E)$-module is projective, and the matrix ring $M_n(L(E))$ is strongly Baer for every $n$. It may not be surprising that a noetherian Leavitt path algebra has these properties, but a more interesting fact is that these properties hold only if a Leavitt path algebra is noetherian. Using some of these equivalences, we give a specific description of the inverse of the isomorphism $V(L(E))\rightarrow V(Q(E))$ of monoids of equivalence classes of finitely generated projective modules for noetherian Leavitt path algebras. We also prove that two noetherian Leavitt path algebras are isomorphic as rings if and only if they are isomorphic as $^\ast$-algebras. This answers in affirmative the Isomorphism Conjecture for the class of noetherian Leavitt path algebras: if $L_{\mathbb C}(E)$ and $L_{\mathbb C}(F)$ are noetherian Leavitt path algebras, then $L_{\mathbb C}(E)\cong L_{\mathbb C}(F)$ as rings implies $C^*(E)\cong C^*(F)$ as $^\ast$-algebras.

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arXiv2013

Classes of almost clean rings

A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite $AW^*$-algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring $R$ is special clean (special almost clean) if each element $a$ can be decomposed as the sum of a unit (regular element) $u$ and an idempotent $e$ with $aR\cap eR=0.$ The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart. If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasi-continuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean. Finally, we adapt some of our results to rings with involution: a *-ring is *-clean (almost *-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) *-clean ring is similarly defined by replacing ``idempotent'' with ``projection'' in the appropriate definition. We show that an abelian *-ring is a Rickart *-ring if and only if it is special almost *-clean, and that an abelian *-ring is *-regular if and only if it is special *-clean.

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arXiv2013

*-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 $^\ast$-regular (i.e. regular with proper involution). This characterization of $^\ast$-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 graph-theoretic properties of $E$ alone. As a corollary, we show that Handelman's conjecture (stating that every $^\ast$-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.

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arXiv2013

Strongly semihereditary rings and rings with dimension

The existence of a well-behaved dimension of a finite von Neumann algebra (see [19]) has lead to the study of such a dimension of finite Baer *-rings (see [26]) that satisfy certain *-ring axioms (used in [9]). This dimension is closely related to the equivalence relation $\sim^{*}$ on projections defined by $p\sim^{*} q$ iff $p=xx^*$ and $q=x^*x$ for some $x.$ However, the equivalence $\sim^{a}$ on projections (or, in general, idempotents) defined by $p\sim^{a} q$ iff $p=xy$ and $q=yx$ for some $x$ and $y,$ can also be relevant. There were attempts to unify the two approaches (see [10]). In this work, our agenda is three-fold: (1) We study assumptions on a ring with involution that guarantee the existence of a well-behaved dimension defined for any general equivalence relation on projections $\sim.$ (2) By interpreting $\sim$ as $\sim^{a},$ we prove the existence of a well-behaved dimension of strongly semihereditary rings with a positive definite involution. This class is wider than the class of finite Baer *-rings with dimension considered in the past: it includes some non Rickart *-rings. Moreover, none of the *-ring axioms from [9] and [26] are assumed. (3) As the first corollary of (2), we obtain dimension of noetherian Leavitt path algebras over positive definite fields. Secondly, we obtain dimension of a Baer *-ring $R$ satisfying the first seven axioms from [26] (in particular, dimension of finite $AW^*$-algebras). Assuming the eight axiom as well, $R$ has dimension for $\sim^{*}$ also and the two dimensions coincide.

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arXiv2011

*-Clean Rings; Some Clean and Almost Clean Baer *-rings and von Neumann Algebras

A ring is clean (resp. almost clean) if each of its elements is the sum of a unit (resp. regular element) and an idempotent. In this paper we define the analogous notion for *-rings: a *-ring is *-clean (resp. almost *-clean) if its every element is the sum of a unit (resp. regular element) and a projection. Although *-clean is a stronger notion than clean, for some *-rings we demonstrate that it is more natural to use. The theorem on cleanness of unit-regular rings from [V. P. Camillo, D. Khurana, A Characterization of Unit Regular Rings, Communications in Algebra, 29 (5) (2001) 2293-2295] is modified for *-cleanness of *-regular rings that are abelian (or reduced or Armendariz). Using this result, it is shown that all finite, type I Baer *-rings that satisfy certain axioms (considered in [S. K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag, Berlin-Heidelberg-New York, 1972] and [L. Vas, Dimension and Torsion Theories for a Class of Baer *-Rings, Journal of Algebra, 289 (2) (2005) 614-639]) are almost *-clean. In particular, we obtain that all finite type I AW*-algebras (thus all finite type I von Neumann algebras as well) are almost *-clean. We also prove that for a Baer *-ring satisfying the same axioms, the following properties are equivalent: regular, unit-regular, left (right) morphic and left (right) quasi-morphic. If such a ring is finite and type I, it is *-clean. Finally, we present some examples related to group von Neumann algebras and list some open problems.

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arXiv2010

A note on $(α, β)$-higher derivations and their extensions to modules of quotients

We extend some recent results on the differentiability of torsion theories. In particular, we generalize the concept of $(α, β)$-derivation to $(α, β)$-higher derivation and demonstrate that a filter of a hereditary torsion theory that is invariant for $α$ and $β$ is $(α, β)$-higher derivation invariant. As a consequence, any higher derivation can be extended from a module to its module of quotients. Then, we show that any higher derivation extended to a module of quotients extends also to a module of quotients with respect to a larger torsion theory in such a way that these extensions agree. We also demonstrate these results hold for symmetric filters as well. We finish the paper with answers to two questions posed in [L. Va\s, Extending higher derivations to rings and modules of quotients, International Journal of Algebra, 2 (15) (2008), 711--731]. In particular, we present an example of a non-hereditary torsion theory that is not differential.

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arXiv2009

Extending ring derivations to right and symmetric rings and modules of quotients

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and perfect torsion theories are differential. We also study conditions under which a derivation on a right or symmetric module of quotients extends to a right or symmetric module of quotients with respect to a larger torsion theory. Using these results, we study extensions of ring derivations to maximal, total and perfect right and symmetric rings of quotients.

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arXiv2009

Perfect Symmetric Rings of Quotients

Perfect Gabriel filters of right ideals and their corresponding right rings of quotients have the desirable feature that every module of quotients is determined solely by the right ring of quotients. On the other hand, symmetric rings of quotients have a symmetry that mimics the commutative case. In this paper, we study rings of quotients that combine these two desirable properties. We define the symmetric versions of a right perfect ring of quotients and a right perfect Gabriel filter -- the perfect symmetric ring of quotients and the perfect symmetric Gabriel filter and study their properties. Then we prove that the standard construction of the total right ring of quotients can be adapted to the construction of the largest perfect symmetric ring of quotients -- the total symmetric ring of quotients. We also demonstrate that Morita's construction of the total right ring of quotients can be adapted to the construction of the total symmetric ring of quotients.

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arXiv2008

Extending higher derivations to rings and modules of quotients

A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsion theories are higher differential as well. Then, we study conditions under which a higher derivation extended to a right module of quotients extends also to a right module of quotients with respect to a larger torsion theory. Lastly, we define and study the symmetric version of higher differential torsion theories. We prove that the symmetric versions of the results on higher differential (one-sided) torsion theories hold for higher derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and any perfect torsion theories are higher differential.

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arXiv2007

Semisimplicity and global dimension of a finite von Neumann algebra

We prove that a finite von Neumann algebra ${\mathcal A}$ is semisimple if the algebra of affiliated operators ${\mathcal U}$ of ${\mathcal A}$ is semisimple. When ${\mathcal A}$ is not semisimple, we give the upper and lower bounds for the global dimensions of ${\mathcal A}$ and ${\mathcal U}.$ This last result requires the use of the Continuum Hypothesis.

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arXiv2007

Differentiability of torsion theories

We prove that every perfect torsion theory for a ring $R$ is differential (in the sense of [P. E. Bland, Differential torsion theory, Journal of Pure and Applied Algebra 204 (2006) 1 -- 8]). In this case, we construct the extension of a derivation of a right $R$-module $M$ to a derivation of the module of quotients of $M$. Then, we prove that the Lambek and Goldie torsion theories for any $R$ are differential.

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arXiv2007

Dimension and Torsion Theories for a Class of Baer *-Rings

Many known results on finite von Neumann algebras are generalized, by purely algebraic proofs, to a certain class ${\mathcal C}$ of finite Baer *-rings. The results in this paper can also be viewed as a study of the properties of Baer *-rings in the class ${\mathcal C}$. First, we show that a finitely generated module over a ring from the class ${\mathcal C}$ splits as a direct sum of a finitely generated projective module and a certain torsion module. Then, we define the dimension of any module over a ring from ${\mathcal C}$ and prove that this dimension has all the nice properties of the dimension studied in [W. Lück, Dimension theory of arbitrary modules over finite von Neumann algebras and $L^2$-Betti numbers I: Foundations, J. Reine Angew. Math. 495 (1998) 135--162] for finite von Neumann algebras. This dimension defines a torsion theory that we prove to be equal to the Goldie and Lambek torsion theories. Moreover, every finitely generated module splits in this torsion theory. If $R$ is a ring in ${\mathcal C},$ we can embed it in a canonical way into a regular ring $Q$ also in ${\mathcal C}.$ We show that $K_0(R)$ is isomorphic to $K_0(Q)$ by producing an explicit isomorphism and its inverse of monoids Proj$ (P)\to$ Proj$ (Q)$ that extends to the isomorphism of $K_0(R)$ and $K_0(Q)$.

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arXiv2007

Class of Baer *-rings Defined by a Relaxed Set of Axioms

We consider a class ${\mathcal C}$ of Baer *-rings (also treated in [S. K. Berberian, Baer *-rings, Die Grundlehren der mathematischen Wissenschaften 195, Springer-Verlag, Berlin-Heidelberg-New York, 1972.] and [L. Vaš, Dimension and Torsion Theories for a Class of Baer *-Rings, Journal of Algebra 289 (2005) no. 2, 614--639]) defined by nine axioms, the last two of which are particularly strong. We prove that the ninth axiom follows from the first seven. This gives an affirmative answer to the question of S. K. Berberian if a Baer *-ring $R$ satisfies the first seven axioms, is the matrix ring $M_n(R)$ a Baer *-ring.

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arXiv2007

A simplification of Morita's construction of total right rings of quotients for a class of rings

The total right ring of quotients $Q_{\mathrm{tot}}^r(R),$ sometimes also called the maximal flat epimorphic right ring of quotients or right flat epimorphic hull, is usually obtained as a directed union of a certain family of extension of the base ring $R$. In [K. Morita, Flat modules, injective modules and quotient rings, Math. Z. 120 (1971) 25--40], $Q_{\mathrm{tot}}^r(R)$ is constructed in a different way, by transfinite induction on ordinals. Starting with the maximal right ring of quotients $Q_{\mathrm{max}}^r(R)$, its subrings are constructed until $Q_{\mathrm{tot}}^r(R)$ is obtained. Here, we prove that Morita's construction of $Q_{\mathrm{tot}}^r(R)$ can be simplified for rings satisfying condition (C) that every subring of the maximal right ring of quotients $Q^r_{\mathrm{max}}(R)$ containing $R$ is flat as a left $R$-module. We illustrate the usefulness of this simplification by considering the class of right semihereditary rings all of which satisfy condition (C). We prove that the construction stops after just one step and we obtain a simple description of $Q^r_{\mathrm{tot}}(R)$ in this case. Lastly, we study conditions that imply that Morita's construction ends in countably many steps.

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arXiv2007

Torsion theories for finite von Neumann algebras

The study of modules over a finite von Neumann algebra ${\mathcal A}$ can be advanced by the use of torsion theories. In this work, some torsion theories for ${\mathcal A}$ are presented, compared and studied. In particular, we prove that the torsion theory $(\mathrm{\bf T},\mathrm{\bf P})$ (in which a module is torsion if it is zero-dimensional) is equal to both Lambek and Goldie torsion theories for ${\mathcal A}$. Using torsion theories, we describe the injective envelope of a finitely generated projective ${\mathcal A}$-module and the inverse of the isomorphism $K_0({\mathcal A})\to K_0({\mathcal U}),$ where ${\mathcal U}$ is the algebra of affiliated operators of ${\mathcal A}.$ Then, the formula for computing the capacity of a finitely generated module is obtained. Lastly, we study the behavior of the torsion and torsion-free classes when passing from a subalgebra ${\mathcal B}$ of a finite von Neumann algebra ${\mathcal A}$ to ${\mathcal A}$. With these results, we prove that the capacity is invariant under the induction of a ${\mathcal B}$-module.

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arXiv2007

Torsion theories for algebras of affiliated operators of finite von Neumann algebras

The dimension of any module over an algebra of affiliated operators ${\mathcal U}$ of a finite von Neumann algebra ${\mathcal A}$ is defined using a trace on ${\mathcal A}.$ All zero-dimensional ${\mathcal U}$-modules constitute the torsion class of torsion theory $(\mathrm{\bf T},\mathrm{\bf P})$. We show that every finitely generated ${\mathcal U}$-module splits as the direct sum of torsion and torsion-free part. Moreover, we prove that the theory $(\mathrm{\bf T},\mathrm{\bf P})$ coincides with the theory of bounded and unbounded modules and also with the Lambek and Goldie torsion theories. Lastly, we use the introduced torsion theories to give the necessary and sufficient conditions for ${\mathcal U}$ to be semisimple.

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