We say that M is a graded R-module if there exists a family of submodules {M g }g∈G of M such that M = ⊕g∈GM g and R g M h ⊆ M gh for all g, h ∈ G, and we write h(M) = ∪g∈GM g.
A module- additive mapping is called a generalized module- left derivation (resp., generalized module- derivation) if there exists a module- left derivation (resp., module- derivation) such that (17).
Under a given set of operating conditions, there exists a module length (named as optimum module length) where the net thermal energy consumption and overall permeate productivity are optimum.
Also, the next lemma shows that there exists a module of finite uniform dimension without couniserial dimension.
Additionally: 1. L K ( m, n ) is universal, in the sense that if S is any K -algebra having module type ( m, n ), then there exists a nonzero K -algebra homomorphism φ : L K ( m, n ) → S. 2.
L K ( m, n ) is universal, in the sense that if S is any K -algebra having module type ( m, n ), then there exists a nonzero K -algebra homomorphism φ : L K ( m, n ) → S. L K ( m, n ) is simple ( i.e., has no nontrivial two-sided ideals ) if and only if m = 1.
It suffices to show that if ϕ ∈ F r Hom M o d ∞ (M, N ), then there exists a module K and homomorphisms π : M → K, ψ : K → N such that π ∈ F 1, ψ ∈ F r − 1 and ϕ = ψ ∘ π.
Recall that a QTAG-module M is (ω+1 -projective if there exists submodule N⊂H1(M) such that M/N is a direct sum of uniserial modules and a QTAG module M is (ω+1 -projective if there exists submodule N⊂H1k (M) such that M/N is a direct sum of uniserial modules [5].
3- It is assumed that there exists another location awareness module not implemented here, this module provides us with some important parameters like: A.
Since (M) is finitely generated, there exists a right module (L) such that for some (k), (E(M) ^{k} cong M oplus L ).
