Definition:Ring of Sequences of Finite Support

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $P \left[{R}\right]$ be the set of all sequences in $R$ whose domain is $\N$:
 * $P \left[{R}\right] = \left\{{\left \langle {r_0, r_1, r_2, \ldots}\right \rangle}\right\}$

and whose support is finite.

Let the operations $\oplus$ and $\odot$ on $P \left[{R}\right]$ be defined as follows:

The ring of sequences of finite support over $R$ is the ring $\left({P \left[{R}\right], \oplus, \odot}\right)$.

Also known as
Because the ring of sequences of finite support can be used to contruct 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:


 * $\forall f = \left \langle {a_k}\right \rangle = \left({a_0, a_1, a_2, \ldots}\right), g = \left \langle {b_k}\right \rangle = \left({b_0, b_1, b_2, \ldots}\right)$:
 * $f \oplus g := \left({a_0 + b_0, a_1 + b_1, a_2 + b_2, \ldots}\right)$
 * $f \otimes g := \left({c_0, c_1, c_2, \ldots}\right)$ 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 $\left({R, +, \circ}\right)$.

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

Also see

 * Polynomial Ring of Sequences is Ring
 * Definition:Polynomial Ring