Definition:Ring of Sequences of Finite Support

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Definition

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

Let $P \sqbrk R$ be the set of all sequences in $R$ whose domain is $\N$:

$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$

and whose support is finite.

Let the operations $\oplus$ and $\odot$ on $P \sqbrk R$ be defined as follows:

\((1)\)   $:$   Ring Addition:       \(\displaystyle \left \langle {r_0, r_1, r_2, \ldots}\right \rangle \oplus \left \langle {s_0, s_1, s_2, \ldots}\right \rangle \)   \(\displaystyle = \)   \(\displaystyle \left \langle {r_0 + s_0, r_1 + s_1, r_2 + s_2, \ldots}\right \rangle \)             
\((2)\)   $:$   Ring Negative:       \(\displaystyle -\left \langle {r_0, r_1, r_2, \ldots}\right \rangle \)   \(\displaystyle = \)   \(\displaystyle \left \langle {-r_0, -r_1, -r_2, \ldots}\right \rangle \)             
\((3)\)   $:$   Ring Product:       \(\displaystyle \left \langle {r_0, r_1, r_2, \ldots}\right \rangle \odot \left \langle {s_0, s_1, s_2, \ldots}\right \rangle \)   \(\displaystyle = \)   \(\displaystyle \left \langle {t_0, t_1, t_2, \ldots}\right \rangle \)             where $\displaystyle t_i = \sum_{j \mathop + k \mathop = i} r_j \circ s_k$


The ring of sequences of finite support over $R$ is the ring $\struct {P \sqbrk R, \oplus, \odot}$.


Also known as

Because the ring of sequences of finite support can be used to construct the polynomial ring over $R$, it may be referred to as a polynomial ring.


Also defined as

Some sources require the operations of polynomial addition and polynomial multiplication to be defined as:

$\forall f = \sequence {a_k} = \tuple {a_0, a_1, a_2, \ldots}, g = \sequence {b_k} = \tuple {b_0, b_1, b_2, \ldots}$:
$f \oplus g := \tuple {a_0 + b_0, a_1 + b_1, a_2 + b_2, \ldots}$
$f \otimes g := \tuple {c_0, c_1, c_2, \ldots}$ where $\displaystyle c_i = \sum_{j \mathop + k \mathop = i} a_j b_k$

before accepting that such a sequence is actually a polynomial.

The argument is that no structure can be imputed to a set of such sequences until these operations are defined upon it.


Also denoted as

It is usual, on presentation of a polynomial ring such as this, to denote the operations $\oplus$ and $\odot$ as the same as those of their counterparts in the underlying ring $\struct {R, +, \circ}$.

However, as this stage of the development of the concepts it is wise to provide separate symbols.


Also see


Sources

Strictly speaking of course, the sequences described are maps of a certain kind from the set $\set {0, 1, \ldots}$ to $R$. However, we prefer to avoid this technically correct notation, fearing it might only conceal the truth from the unfamiliar eye, and leave those well versed in such symbolic punctiliousness to make the translation themselves.