Definition of a multilinear polynomial Is a multilinear polynomial of variables $x_1, \dots, x_n$ over a ring defined as a monomial $c \prod_{i=1}^n x_i$, where $c$ is a constant from the ring?
Equivalently, is a multilinear polynomial function of variables $x_1, \dots, x_n$ over a ring  same as a multilinear mapping of $x_1, \dots, x_n$?
My confusion comes from Wikipedia

In algebra, a multilinear polynomial is a polynomial that is linear in
  each of its variables. In other words, no variable occurs to a power
  of 2 or higher; or alternatively, each monomial is a constant times a
  product of distinct variables. ... The degree of a multilinear
  polynomial is the maximum number of distinct variables occurring in
  any monomial.

It seems to suggest a multilinear polynomial can have more than one monomial terms. Thanks!
 A: According to this definition, as Arturo said, a multilinear polynomial in $k[x_1,x_2,\ldots,x_n]$ is not the same as a polynomial that induces a multilinear mapping on $k^n$, the latter being only $c\prod\limits_{k=1}^n x_k$ as you indicated.  There aren't many interesting multilinear maps from $k^n\to k$ when $k$ is a field, but the multilinear polynomials according to this definition include any polynomial such that each monomial is squarefree.  This means that fixing $n-1$ of the variables induces an affine map $k\to k$, i.e. a map of the form $x\mapsto ax+b$.  
On the other hand, if $R$ is a noncommutative ring, then there may be more interesting multilinear maps $R^n\to R$, and consequently reason for restricting the definition of multilinear polynomial to mean one that induces a multilinear map.  But in this case the polynomials should be in noncommuting variables.  E.g., if $R$ is a $k$-algebra, then elements of the "noncommutative polynomial ring" (i.e., free algebra) $k\langle x_1,x_2,\ldots,x_n\rangle$ induce maps $R^n\to R$, and include nonmonomial multilinear polynomials (in the restricted sense) like $x_1x_2\cdots x_n + x_n x_{n-1}\cdots x_1$.  Such (noncommutative) multilinear polynomials are important in the theory of polynomial identity rings.
