Definition:Zero Divisor/Algebra

Definition
Let $\left({A_R, \oplus}\right)$ be an algebra over a ring $\left({R, +, \cdot}\right)$.

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 $A_R$ iff $a \oplus b = \mathbf 0_R$.