The word product  in mathematics generally means the result of some type of multiplication operation, i.e. of certain types of mapping  $X\!\times\!X\rightarrow Y$; such operations are commonly distributive over the addition operation of $X$ if it is defined.

If $x_{1}$ and $x_{2}$ are two elements of the set $X$, giving the product  $y\in Y$, then $x_{1}$ and $x_{2}$ are in general called the factors of this product.

Some most usual products are

• the ring product (also in fields), especially the product of numbers and the product of square matrices;

• on vectors the scalar product, the vector product, the dyad product and the Hadamard product; on ideals the product of ideals;

• the Cartesian product, the direct products of various systems (not in connection with any additions).
  

Such kinds of product that are associative, allow to form a product of more than two factors, which is justified by the theorem in the entry general associativity.  E.g. the usual product of the integers from 1 to $n$ is the factorial of $n$.