How to understand Free Module $K[x]^r=\bigoplus_{i=r}^rK[x]e_i$

Question

How can i understand of the free module $K[x]^r=\bigoplus_{i=r}^rK[x]e_i$ where $e_i=(0,\ldots ,1, \ldots 0) \in K[x]^r $ denotes the i–th canonical basis vector of $K[x]^r$. We call $x^\alpha e_i=(0, \ldots , x^\alpha , \ldots 0)$ a monomial. For a monomial $x^\alpha e_i \in K[x]^r$ set $\operatorname{deg} x^\alpha e_i:=\operatorname{deg} x^\alpha=\alpha_1+\cdots+\alpha_n$ But im confused because i think $\alpha$ is a monomial just can be a element of $\mathbb{Z}$ because we are working with polynomials in just one variable?can you explain me? any example?

Answer 1

It is a common shorthand in commutative algebra to write $K[x]$ to mean polynomials in a finite number of variables $x_1,\dots, x_n$ with coefficients in a field $K$. As you note, this is a slight abuse of notation, since $K[x]$ does look like it's one variable. Some authors try to disambiguate with notations like $K[\mathbf{x}]$ or $K[X]$, but others do not.

As noted in the comments, this question arises from reading Greuel and Pfister's A Singular Introduction to Commutative Algebra. In the edition I am referencing, this material looks like it appears in Section 2.3, "Standard Bases for Modules." From what I can see, the authors have up to this point freely used the notation $K[x]$ for both single-variable and the $n$-variable polynomial rings. See for instance the subalgebra membership problem in 1.8.11 as well as various exercises in that section.

Therefore, we should interpret $\alpha$ as an element of $\mathbb{N}^n$ (where $0\in\mathbb{N}$) and then $x^\alpha$ in the usual manner, e.g. $x^{(3,1,0,0,2)}=x_1^3x_2x_5^2$. Note that in general this $n$ is the number of variables in the ring, and does not need to be equal to the rank of the free module $r$; the two "vectors" appearing in the notation $x^\alpha e_i$ are unrelated.