Definition:Matrix Product (Conventional)/Einstein Summation Convention

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $\mathbf A = \left[{a}\right]_{m n}$ be an $m \times n$ matrix over $R$.

Let $\mathbf B = \left[{b}\right]_{n p}$ be an $n \times p$ matrix over $R$.

The matrix product of $\mathbf A$ and $\mathbf B$ can be expressed using the summation convention as:

Then:
 * $c_{i j} := a_{i k} \circ b_{k j}$

The index which appears twice in the expressions on the RHS is the element $k$, which is the one summated over.