Definition:Ring of Sequences

Definition
Let $\struct {R, +, \circ}$ be a non-null ring.

Given the Natural numbers $\N$, the ring of sequences over $R$ is the ring $\struct {R^{\N}, +', \circ'}$ induced on $R^{\N}$ by $+$ and $\circ$.

That is, the ring of sequences is the set of sequences in R with pointwise addition and pointwise product.

The ring operations on the ring of sequences over $R$ are defined:
 * $\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$
 * $\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$

The zero of the ring of sequences is the sequence $\tuple{0,0,0,\dots}$, where $0$ is the zero in $R$.

By Structure Induced by Ring with Unity is a Ring with Unity, if $R$ is a ring with unity then the ring of sequences over $R$ is a ring with unity; namely the sequence $\tuple{1,1,1,\dots}$, where $1$ is the unity in $R$.

By Structure Induced by Commutative Ring is a Commutative Ring, if $R$ is a commutative ring then the ring of sequences over $R$ is a commutative ring.

Also denoted as
It is usual to use the same symbols for the induced operations on the ring of sequences over $R$ as for the operations that induces them.

Also see

 * Structure Induced by Ring Operations is Ring


 * Structure Induced by Ring with Unity Operations is Ring with Unity


 * Structure Induced by Commutative Ring Operations is Commutative Ring