Definition:Product (Abstract Algebra)

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Definition

General Operation

Let $z = x \circ y$.

Then $z$ is called the product of $x$ and $y$.

This is an extension of the normal definition of product that is encountered in conventional arithmetic.


Group Product

The operation $\circ$ can be referred to as the group law.


Ring Product

The distributive operation $\circ$ in $\struct {R, *, \circ}$ is known as the (ring) product.


Field Product

The distributive operation $\times$ in $\struct {F, +, \times}$ is known as the (field) product.