Definition:Zero Divisor/Algebra

Definition
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 $b \ne \mathbf 0_R$.

Then $a$ is a zero divisor of $A_R$ :
 * $a \oplus b = \mathbf 0_R$