Definition:Ring of Sequences
Definition
Let $\struct {R, +, \circ}$ be a ring.
Given the natural numbers $\N$, the ring of sequences over $R$ is the ring of mappings $\struct {R^\N, +', \circ'}$ where:
- $(1): \quad R^\N$ is the set of sequences in $R$
- $(2): \quad +'$ and $\circ'$ are the (pointwise) operations induced by $+$ and $\circ$.
From Structure Induced by Ring Operations is Ring, $\struct {R^\N, +', \circ'}$ is a ring.
Pointwise Addition
The pointwise operation $+'$ induced by $+$ on the ring of sequences is called pointwise addition and is defined as:
- $\forall \sequence {x_n}, \sequence {y_n} \in R^\N: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$
Pointwise Multiplication
The pointwise operation $\circ'$ induced by $\circ$ on the ring of sequences is called pointwise multiplication and is defined as:
- $\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$
Zero of Ring of Sequences
The zero of the ring of sequences is the constant sequence $\tuple {0, 0, 0, \dots}$, where $0$ is the zero of $R$.
Additive Inverse
The additive inverse in the ring of sequences is defined by:
- $\forall \sequence {x_n} \in R^\N: -\sequence {x_n} = \sequence {-x_n}$
Unity of Ring of Sequences
Let $\struct {R, +, \circ}$ be a ring with unity $1$.
From Structure Induced by Ring with Unity Operations is Ring with Unity, the ring of sequences is a ring with unity; namely the constant sequence $\tuple {1, 1, 1, \dots}$, where $1$ is the unity in $R$.
Units of Ring of Sequences with Unity
Let $\struct {R, +, \circ}$ be a ring with unity $1$.
Let $\sequence {x_n}$ be a sequence over the set of units $U_R$ of $R$.
From Unit of Ring of Mappings iff Image is Subset of Ring Units:
- $\sequence {x_n}$ is a unit in the ring of sequences over $R$
and:
- the inverse of $\sequence {x_n}$ is the sequence defined by:
- $\sequence {x_n}^{-1} \in R^\N : \sequence {x_n}^{-1} = \sequence {x_n^{-1}}$
Commutativity of Ring of Sequences
Let $\struct {R, +, \circ}$ be a commutative ring.
From Structure Induced by Commutative Ring Operations is Commutative Ring, 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
- Results about rings of sequences can be found here.
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