Salvatore Tringali - Academia.edu (original) (raw)
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Papers by Salvatore Tringali
We combine the language of monoids with the language of preorders to formulate an abstract factor... more We combine the language of monoids with the language of preorders to formulate an abstract factorization theorem with several applications. In particular, this leads to (i) a generalization of P.M. Cohn’s classical theorem on “atomic factorizations” from cancellative to Dedekind-finite monoids (and, hence, to a variety of rings that are not domains); (ii) a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring R is isomorphic to a direct sum of finitely many indecomposable R-modules (in fact, we obtain the result as a special case of a general decomposition theorem for the objects of certain categories with finite products, where the indecomposable R-modules are characterized as the atoms of a certain “monoid of modules”). Also, we recover and extend an existence theorem of D.D. Anderson and S. Valdes-Leon on “irreducible factorizations” in commutative rings [RMJM 1996]; a refinement of Cohn’s theorem to “nearly cancellativ...
Given an integer n ≥ 3, let u 1 , . . . , u n be pairwise coprime integers for which 2 ≤ u 1 < · ... more Given an integer n ≥ 3, let u 1 , . . . , u n be pairwise coprime integers for which 2 ≤ u 1 < · · · < u n , and let D be a family of nonempty proper subsets of {1, . . . , n} with "enough" elements and ε a map D → {±1}. Does there exist at least one q ∈ P such that q divides i∈I u i − ε(I) for some I ∈ D and q u 1 · · · u n ? We answer this question in the positive in the case where the integers u i are prime powers and some restrictions hold on ε and D. We use the result to prove that, if ε 0 ∈ {±1} and A is a set of three or more primes with the property that A contains all prime divisors of any product of the form p∈B p − ε 0 for which B is a finite nonempty proper subset of A, then A contains all the primes.
2009 International Conference on Electromagnetics in Advanced Applications, 2009
It is well established in literature that the rate of convergence of the Generalized Minimum Resi... more It is well established in literature that the rate of convergence of the Generalized Minimum Residual Method (GMRES), when it is applied to the solution of the (generally dense and unstructured) linear systems of equations coming out from the discretization process of the Electrical Field Integral Equation (EFIE) through the Method of Moments (MoM), can be significantly improved by a
Let A = (A, ·) be a semigroup. We generalize some recent results by G. Freiman, M. Herzog and coa... more Let A = (A, ·) be a semigroup. We generalize some recent results by G. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that A is linearly orderable if there exists a total order ≤ on A such that xz < yz and zx < zy for all x, y, z ∈ A with x < y.
We combine the language of monoids with the language of preorders to formulate an abstract factor... more We combine the language of monoids with the language of preorders to formulate an abstract factorization theorem with several applications. In particular, this leads to (i) a generalization of P.M. Cohn’s classical theorem on “atomic factorizations” from cancellative to Dedekind-finite monoids (and, hence, to a variety of rings that are not domains); (ii) a monoid-theoretic proof that every module of finite uniform dimension over a (commutative or non-commutative) ring R is isomorphic to a direct sum of finitely many indecomposable R-modules (in fact, we obtain the result as a special case of a general decomposition theorem for the objects of certain categories with finite products, where the indecomposable R-modules are characterized as the atoms of a certain “monoid of modules”). Also, we recover and extend an existence theorem of D.D. Anderson and S. Valdes-Leon on “irreducible factorizations” in commutative rings [RMJM 1996]; a refinement of Cohn’s theorem to “nearly cancellativ...
Given an integer n ≥ 3, let u 1 , . . . , u n be pairwise coprime integers for which 2 ≤ u 1 < · ... more Given an integer n ≥ 3, let u 1 , . . . , u n be pairwise coprime integers for which 2 ≤ u 1 < · · · < u n , and let D be a family of nonempty proper subsets of {1, . . . , n} with "enough" elements and ε a map D → {±1}. Does there exist at least one q ∈ P such that q divides i∈I u i − ε(I) for some I ∈ D and q u 1 · · · u n ? We answer this question in the positive in the case where the integers u i are prime powers and some restrictions hold on ε and D. We use the result to prove that, if ε 0 ∈ {±1} and A is a set of three or more primes with the property that A contains all prime divisors of any product of the form p∈B p − ε 0 for which B is a finite nonempty proper subset of A, then A contains all the primes.
2009 International Conference on Electromagnetics in Advanced Applications, 2009
It is well established in literature that the rate of convergence of the Generalized Minimum Resi... more It is well established in literature that the rate of convergence of the Generalized Minimum Residual Method (GMRES), when it is applied to the solution of the (generally dense and unstructured) linear systems of equations coming out from the discretization process of the Electrical Field Integral Equation (EFIE) through the Method of Moments (MoM), can be significantly improved by a
Let A = (A, ·) be a semigroup. We generalize some recent results by G. Freiman, M. Herzog and coa... more Let A = (A, ·) be a semigroup. We generalize some recent results by G. Freiman, M. Herzog and coauthors on the structure theory of set addition from the context of linearly orderable groups to linearly orderable semigroups, where we say that A is linearly orderable if there exists a total order ≤ on A such that xz < yz and zx < zy for all x, y, z ∈ A with x < y.