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Given a module M over a domestic canonical algebra Λ and a classifying set X for the indecomposable Λ-modules, the problem of determining the vector $m (M) = {(m_{x})}_{x \in X} \in ℕ^{X}$ such that $M ≅ ⨁_{x \in X} X_{x}^{m_{x}}$ is studied. A precise formula for $d i m_{k} H o m_{Λ} (M, X)$ , for any postprojective indecomposable module X, is computed in Theorem 2.3, and interrelations between various structures on the set of all postprojective roots are described in Theorem 2.4. It is proved in Theorem 2.2 that a general method of finding vectors m(M) presented by the authors in Colloq....

The multiplicity problem for indecomposable decompositions of modules over a finite-dimensional algebra. Algorithms and a computer algebra approach

Piotr Dowbor, Andrzej Mróz (2007)

Colloquium Mathematicae

Given a module M over an algebra Λ and a complete set of pairwise nonisomorphic indecomposable Λ-modules, the problem of determining the vector $m (M) = {(m_{X})}_{X \in} \in ℕ^{}$ such that $M ≅ ⨁_{X \in} X^{m_{X}}$ is studied. A general method of finding the vectors m(M) is presented (Corollary 2.1, Theorem 2.2 and Corollary 2.3). It is discussed and applied in practice for two classes of algebras: string algebras of finite representation type and hereditary algebras of type $̃_{p, q}$ . In the second case detailed algorithms are given (Algorithms 4.5 and 5.5).

The multipliers of similitudes and the Brauer group of homogeneous varieties.

J.-P. Tignol, A.S. Merkurjev (1995)

Journal für die reine und angewandte Mathematik

The multisegment duality and the preprojective algebras of type $A$ .

Ringel, Claus Michael (1999)

AMA. Algebra Montpellier Announcements [electronic only]

The near-ring of Lipschitz functions on a metric space.

Farag, Mark, Van Der Merwe, Brink (2010)

International Journal of Mathematics and Mathematical Sciences

The Near-rings on Groups of Low Order.

James R. Clay (1968)

Mathematische Zeitschrift

The nil radical of an Archimedean partially ordered ring with positive squares

Boris Lavrič (1994)

Commentationes Mathematicae Universitatis Carolinae

Let $R$ be an Archimedean partially ordered ring in which the square of every element is positive, and $N (R)$ the set of all nilpotent elements of $R$ . It is shown that $N (R)$ is the unique nil radical of $R$ , and that $N (R)$ is locally nilpotent and even nilpotent with exponent at most $3$ when $R$ is 2-torsion-free. $R$ is without non-zero nilpotents if and only if it is 2-torsion-free and has zero annihilator. The results are applied on partially ordered rings in which every element $a$ is expressed as $a = a_{1} - a_{2}$ with positive $a_{1}$ ,...

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