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ABSTRACT: Hazrat gave a K-Theoretic invariant for Leavitt path algebras as graded
algebras. Hazrat conjectured that this invariant classifies Leavitt path
algebras up to graded isomorphism, and proved the conjecture in some cases. In
this paper, we prove that the conjecture holds for all finite graphs with
neither sources nor sinks.
10/2012;
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ABSTRACT: In this article we prove that if S is an irreducible numerical semigroup and S is generated by an interval or S has multiplicity 3 or 4, then it enjoys Toms decomposition. We also prove that if a numerical semigroup can be expressed as an expansion of a numerical semigroup generated by an interval, then it is irreducible and has Toms decomposition.
Communications in Algebra 02/2007; 35(2):501-513. · 0.35 Impact Factor
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ABSTRACT: We characterize the values of the stable rank for Leavitt path algebras, by giving concrete criteria in terms of properties of the underlying graph.
10/2006;
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ABSTRACT: We compute the monoid $V(L_K(E))$ of isomorphism classes of finitely generated projective modules over certain graph algebras $L_K(E)$, and we show that this monoid satisfies the refinement property and separative cancellation. We also show that there is a natural isomorphism between the lattice of graded ideals of $L_K(E)$ and the lattice of order-ideals of $V(L_K(E))$. When $K$ is the field $\mathbb C$ of complex numbers, the algebra $L_{\mathbb C}(E)$ is a dense subalgebra of the graph $C^*$-algebra $C^*(E)$, and we show that the inclusion map induces an isomorphism between the corresponding monoids. As a consequence, the graph C*-algebra of any row-finite graph turns out to satisfy the stable weak cancellation property.
01/2005;
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ABSTRACT: Given an action of a monoid $T$ on a ring $A$ by ring endomorphisms, and an Ore subset $S$ of $T$, a general construction of a fractional skew monoid ring $S^{\rm op} * A * T$ is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case $S$ is a subsemigroup of a group $G$ such that $G=S^{-1}S$, we obtain a $G$-graded ring $S^{\rm op} * A * S$ with the property that, for each $s\in S$, the $s$-component contains a left invertible element and the $s^{-1}$-component contains a right invertible element. In the most basic case, where $G$ is the additive group of integers and $S=T$ is the submonoid of nonnegative integers, the construction is fully determined by a single ring endomorphism $\alpha$ of $A$. If $\alpha $ is an isomorphism onto a proper corner $pAp$, we obtain an analogue of the usual skew Laurent polynomial ring, denoted by $A[t_+,t_-;\alpha]$. Examples of this construction are given, and it is proven that several classes of known algebras, including the Leavitt algebras of type $(1,n)$, can be presented in the form $A[t_+,t_-;\alpha]$. Finally, mild and reasonably natural conditions are obtained under which $S^{\rm op} * A * S$ is a purely infinite simple ring.
08/2003;
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ABSTRACT: We extend the notion of a purely infinite simple C*-algebra to the context of unital rings, and we study its basic properties, specially those related to K-Theory. For instance, if $R$ is a purely infinite simple ring, then $K_0(R)^+= K_0(R)$, the monoid of isomorphism classes of finitely generated projective $R$-modules is isomorphic to the monoid obtained from $K_0(R)$ by adjoining a new zero element, and $K_1(R)$ is the abelianization of the group of units of $R$. We develop techniques of construction, obtaining new examples in this class in the case of von Neumann regular rings, and we compute the Grothendieck groups of these examples. In particular, we prove that every countable abelian group is isomorphic to $K_0$ of some purely infinite simple regular ring. Finally, some known examples are analyzed within this framework.
12/2001;
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ABSTRACT: A separative ring is one whose finitely generated projective modules satisfy the propertyA⊕A⋟A⊕B⋟B⊕B⇒A⋟B. This condition is shown to provide a key to a number of outstanding cancellation problems for finitely generated projective
modules over exchange rings. It is shown that the class of separative exchange rings is very broad, and, notably, closed under
extensions of ideals by factor rings. That is, if an exchange ringR has an idealI withI andR/I both separative, thenR is separative.
Israel Journal of Mathematics 04/1998; 105(1):105-137. · 0.75 Impact Factor
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ABSTRACT: Square matrices are shown to be diagonalizable over all known classes of (von Neumann) regular rings. This diagonalizability is equivalent to a cancellation property for finitely generated projective modules which conceivably holds over all regular rings. These results are proved in greater generality, namely for matrices and modules over exchange rings, where attention is restricted to regular matrices.
Linear Algebra and its Applications.
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ABSTRACT: Given an action α of a monoid T on a ring A by ring endomorphisms, and an Ore subset S of T, a general construction of a fractional skew monoid ring is given, extending the usual constructions of skew group rings and of skew semigroup rings. In case S is a subsemigroup of a group G such that G=S−1S, we obtain a G-graded ring with the property that, for each s∈S, the s-component contains a left invertible element and the s−1-component contains a right invertible element. In the most basic case, where and , the construction is fully determined by a single ring endomorphism α of A. If α is an isomorphism onto a proper corner pAp, we obtain an analogue of the usual skew Laurent polynomial ring, denoted by A[t+,t−;α]. Examples of this construction are given, and it is proven that several classes of known algebras, including the Leavitt algebras of type (1,n), can be presented in the form A[t+,t−;α]. Finally, mild and reasonably natural conditions are obtained under which is a purely infinite simple ring.
Journal of Algebra.
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ABSTRACT: We prove that, for every regular ring R, there exists an isomorphism between the monoids of isomorphism classes of finitely generated projective right modules over the rings EndR(R(ω)R) and RCFM(R), where the latter denotes the ring of countably infinite row- and column-finite matrices over R. We use this result to give a precise description of the countably generated projective modules over simple regular rings and over regular rings satisfying s-comparability.
Journal of Algebra. 226(1):161-190.
