Algebra

Abstract

Let F be a field (cf. 4.7), and let L be a set endowed with two operations: the addition of elements of L, and the multiplication of elements of L by elements of F such that (L, +) is a commutative group (i.e., fulfils conditions (2.9.1)–(2.9.4); cf. 4.5), and moreover

$$ 1x = x $$

for every x ∈ L,

$$ \alpha (\beta x) = (\alpha \beta )x $$

for every α, β ∈ F, x ∈ L,

$$ (\alpha + \beta )x = \alpha x + \beta x, \alpha (x + y) = \alpha x + \alpha y $$

for every α, β ∈ F, x, y ∈ L.