# Definition:Product (Abstract Algebra)

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## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\circ$ be the operation on $\struct {S, \circ}$.

### 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

Let $\struct {G, \circ}$ be a group.

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

### Ring Product

Let $\struct {R, *, \circ}$ be a ring.

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

### Field Product

Let $\struct {F, +, \times}$ be a field.

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