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Definition

Let RR be a commutative ring.

Example

The set of polynomials in one variable with coefficients in RR is the set R[]R[\mathbb{N}] of all formal linear combinations on elements nn \in \mathbb{N}, thought of as powers x nx^n of the variable xx

a nz n++a 1z+a 0, a_n z^n + \cdots + a_1 z + a_0 \,,

where nn is an arbitrary natural number and a 0,,a nRa_0, \dots, a_n \in R, modulo the equivalence relation generated by equations of the form

0z n+1+a nz n++a 1z+a 0=a nz n++a 1z+a 0 0 z^{n+1} + a_n z^n + \cdots + a_1 z + a_0 = a_n z^n + \cdots + a_1 z + a_0

(so that we ignore coefficients of zero).

This set is equipped with the structure of a ring itself, in fact a commutative algebra over RR, denoted R[z]R[z] and called the polynomial ring or ring of polynomials given by the unique bilinear map

R[z]R[z]R[z] R[z] \cdot R[z] \to R[z]

which on monomials is given by

z kz l=z k+l. z^k \cdot z^l = z^{k+l} \,.

In addition to the ring structure, there is a further operation R[z]×R[z]R[z]R[z] \times R[z] \to R[z] which may be described as “substitution”; see Remark 2 below for a general description (which applies in fact to any Lawvere theory).

Moreover, there is a noncommutative analogue of polynomial ring on a set XX, efficiently described as the free RR-module generated by the (underlying set of the) free monoid on XX. This carries also a ring structure, with ring multiplication induced from the monoid multiplication. A far-reaching generalization of this construction is given at distributive law.

Finally: polynomial algebras may be regarded as graded algebras (graded over \mathbb{N}). Specifically: let us regard R[X]R[X] as the free RR-module generated by (the underlying set of) the free commutative monoid F(X)F(X). The monoid homomorphism F(!):F(X)F(1)F(!): F(X) \to F(1) \cong \mathbb{N} induced by the unique function !:X1!: X \to 1 gives an \mathbb{N}-fibering of F(X)F(X) over \mathbb{N}, with typical fiber F(X) nF(X)_n whose elements are called monomials of degree nn. Then the homogeneous component of degree nn in R[X]R[X] is the RR-submodule generated by the subset F(X) nF(X)F(X)_n \subset F(X). The elements of this component are called homogeneous polynomials of degree nn.

Properties

Proposition

The polynomial ring R[z]R[z] is the free RR-algebra on one generator (the variable zz).

Proof

By the definition of free objects one needs to check that ring homomorphisms

f:R[z]K f : R[z] \to K

to another ring K are in natural bijection with functions of sets

f¯:*K \bar f : * \to K

from the singleton to the set underlying KK. Take f¯f(z)\bar f \coloneqq f(z). Using RR-linearity, this is directly seen to yield the desired bijection.

Remark

Similarly, the set of polynomials in any give set of variables with coefficients in RR is the free commutative RR-algebra on that set of generators; see symmetric power and symmetric algebra.

Remark

As usual in the study of universal algebra via Lawvere theories, there is an operad whose n thn^{th} component C nC_n is the free algebra R[x 1,,x n]R[x_1, \ldots, x_n], and whose operadic multiplication is given by maps

C n×C n 1××C n kC nC_n \times C_{n_1} \times \ldots \times C_{n_k} \to C_n

(n=n 1++n kn = n_1 + \ldots + n_k) that take a tuple of elements (p;q 1,,q k)(p; q_1, \ldots, q_k) to p(q 1(x),,q k(x))p(q_1(x), \ldots, q_k(x)). Formally, it takes this tuple to the value of pp under the unique algebra map R[x 1,,x k]R[x 1,,x n]R[x_1, \ldots, x_k] \to R[x_1, \ldots, x_n] that extends the mapping x ji j(q j)x_j \mapsto i_j(q_j). Here the i j:R[x 1,,x n j]R[x 1,,x n]i_j: R[x_1, \ldots, x_{n_j}] \to R[x_1, \ldots, x_n] are appropriate coproduct inclusions (in the category of commutative rings), where i j(x l)=x n 1++n j1+li_j(x_l) = x_{n_1 + \ldots + n_{j-1} + l}. A particularly important case of substitution is the case k=1k=1 and n 1=1n_1 = 1, where the map R[x]×R[x]R[x]R[x] \times R[x] \to R[x] is ordinary substitution (p,q)p(q(x))(p, q) \mapsto p(q(x)). This is a special case of the more general notion of Tall-Wraith monoid.

Remark

In case RR is an integral domain, the field of fractions of R[z]R[z] is the field R(z)R(z) of rational functions.

Revised on January 3, 2015 20:22:44 by Todd Trimble (127.0.0.1)