We show that the operation of taking generic extensions provides the set of isomorphism classes of representations of a quiver of Dynkin type with a monoid structure. Its monoid ring is isomorphic to the specialization at q = 0 of Ringel’s Hall algebra. This provides the latter algebra with a multiplicatively closed basis. Using a crystal-type basis for a two-parameter quantum group, this multi...

A module M is called product closed if every hereditary pretorsion class in σ[M ] is closed under products in σ[M ]. Every module which is locally of finite length is product closed and every product closed module is semilocal. LetM ∈ R-Mod be product closed and projective in σ[M ]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invarian...

Let R be a commutative ring, / an ideal in R, and A an i?-module. We always have 0 £= 0 £ I(\~=1 I A £ f|ϊ=i JM. where S is the multiplicatively closed set {1 — i\ie 1} and 0 = 0s Π A = {α G A13S 6 S 3 sα = 0}. It is of interest to know when some containment can be replaced by equality. The Krull intersection theorem states that for R Noetherian and A finitely generated I Π*=i I A = Π~=i IA. Si...

(1) Let S1 be a finite non empty subset of R and e be a real number. If for every real number r such that r ∈ S1 holds r < e, then maxS1 < e. For simplicity, we follow the rules: C is a simple closed curve, A, A1, A2 are subsets of E2 T, p, p1, p2, q, q1, q2 are points of E2 T, and n is a natural number. Let us consider n. Observe that there exists a subset of En T which is trivial. One can pro...

(4.1) Let E,F be nonempty subsets of R, E is compact and F is closed. Then there exist (e, f) ∈ E ×F such that d(E,F ) = d(e, f). proof Let α = d(E,F ) = inf(x,y)∈E×F d(x, y) be the distance between the two sets. Let > 0, and let x0 be a given element of E. Since E is compact, it is in particular bounded, and there exists r > 0 such that E ⊂ B(r, x0). Now consider the closed ball B(r + α+ , x0)...

This paper introduces an implicit scheme for a   continuous representation of nonexpansive mappings on a closed convex subset of a Hilbert space with respect to a   sequence of invariant means defined on an appropriate space of bounded, continuous real valued functions of the semigroup.   The main result is to    prove the strong convergence of the proposed implicit scheme to the unique solutio...

a definition of two jointly asymptotically nonexpansive mappings s and t on uniformly convex banach space e is studied to approximate common fixed points of two such mappings through weak and strong convergence of an ishikawa type iteration scheme generated by s and t on a bounded closed and convex subset of e. as a consequence of the notion of two jointly asymptotically nonexpansive maps, we c...