Université Catholique de Louvain Domains of Submodules, Join and Meet of Finite Sequences.pdf

FORMALIZED MATHEMATICS Volume 3, Number 2, 1992 Universit´e Catholique de Louvain Domains of Submodules, Join and Meet of Finite Sequences of Submodules and Quotient Modules Michal Muzalewski Warsaw University Bialystok Summary. Notions of domains of submodules, join and meet of finite sequences of submodules and quotient modules. A few basic theo rems and schemes related to these notions are proved. MML Identifier: LMOD 7. The papers [17], [28], [3], [4], [2], [1], [16], [5], [29], [15], [24], [20], [25], [27], [21], [18], [7], [6], [8], [26], [23], [22], [19], [14], [13], [11], [12], [9], and [10] provide the terminology and notation for this paper. 1. Auxiliary theorems on freemodules For simplicity we follow a convention: x is arbitrary, K is an associative ring, r is a scalar of K , V , M , N are left modules over K , a, b, a , a are vectors of V , 1 2 A, A , A are subsets of V , l is a linear combination of A, W is a submodule 1 2 of V , and L1 is a finite sequence of elements of Sub(V ). One can prove the following propositions: (1) If K is non-trivial and A is linearly independent, then 0V ∈/ A. (2) If a ∈/ A, then l(a) = 0K . (3) If K is trivial, then for every l holds support l = ∅ and Lin(A) is trivial. (4) If V is non-trivial, then for every A such that A is base holds A = ∅. (5) If A ∪ A is linearly independent and A ∩ A = ∅, then Lin(A ) ∩ 1 2 1 2 1 Lin(A ) = 0 . 2