Is an Ideal of Polynomials Without Constant Terms Finitely Generated?

[mesarmath]

hi,

i was reading about finitely generated ideals

and there was a remark that ideal which consists of polynomials with no constant term, in the polynomial ring Z[x_1,x_2,x_3,,,,,,] , is not finitely generated.

and i can not show it is not finitely generated

any idea?

 

[mrbohn1]

This is only true if the ring R=Z[x_1,x_2,x_3,...] has an infinite number of variables. The ring itself is finitely generated as an R-module by the identity {1}. Clearly we can't use {1} to generate the submodule consisting of all polynomials with no constant term though.

The proof that this submodule is not finitely generated hinges on the fact that polynomials have only finitely many non-zero terms. This means that any finite generating set of polynomials must generate only finitely many variables. The submodule has countably many variables, so cannot be finitely generated.