finitely generated module (original) (raw)

Examples. Let R be a commutative ring with 1 and x be an indeterminate.

1. 1. R⁢x={r⁢x∣r∈R} is a cyclic R-module generated by {x}.
2. 1. R⊕R⁢x is a finitely-generated R-module generated by {1,x}. Any element in R⊕R⁢xcan be expressed uniquely as r+s⁢x.
3. 1. R⁢[x] is not finitely generated as an R-module. For if there is a finite set Y R⁢[x], taking d to be the largest of all degrees of polynomials in Y, then xd+1 would not be in the of Y, assumed to be R⁢[x], which is a contradiction. (Note, however, that R⁢[x] is finitely-generated as an R-algebra.)