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 $$I(v)=\left\{i\in I:a_{i}(v)\neq 0\right\}$$ 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 $0\neq w\in W,$ and
2. $a_{i}(p)=1$ for some index $i.$