# Definition:Zero Divisor/Algebra

Let $\struct {A_R, \oplus}$ be an algebra over a ring $\struct {R, +, \cdot}$.
Let the zero vector of $A_R$ be $\mathbf 0_R$.
Let $a, b \in A_R$ such that $a \ne \mathbf 0_R$ and $b \ne \mathbf 0_R$.
Then $a$ and $b$ are zero divisors of $A_R$ if and only if:
$a \oplus b = \mathbf 0_R$