Primordial element (algebra)

In algebra, a primordial element is a particular kind of a vector in a vector space.

Definition

Let $V$ be a vector space over a field $\mathbb{𝔽}$ and let ${\left(e_i\right)}_{i\in I}$ be an $I$-indexed basis of vectors for $V.$ By the definition of a basis, every vector $v\in V$ can be expressed uniquely as $$v=\sum _{i\in I}{a}_i(v)e_i$$ for some $I$-indexed family of scalars ${\left(a_i\right)}_{i\in I}$ where all but finitely many $a_i$ are zero. Let $$I(v)=\{i\in I:a_i(v)\ne 0\}$$ denote the set of all indices for which the expression of $v$ has a nonzero coefficient. Given a subspace $W$ of $V,$ a nonzero vector $p\in W$ is said to be primordial if it has both of the following two properties:

1. $I(p)$ is minimal among the sets $I(w),$ where $0\ne w\in W,$ and
2. $a_i(p)=1$ for some index $i.$