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.