Let R be a commutative k -algebra, where k is a field of characteristic zero. Coul

Brock Byrd 2022-07-10 Answered
Let R be a commutative k-algebra, where k is a field of characteristic zero.
Could one please give an example of such R which is also:
(i) Not affine (= infinitely generated as a k-algebra).
and
(ii) Not an integral domain (= has zero divisors).My first thought was k [ x 1 , x 2 , ], the polynomial ring over k in infinitely many variables, but unfortunately, it satisfies condition (i) only. It is not difficult to see that it is an integral domain: If f g = 0 for some f , g k [ x 1 , x 2 , ], then there exists M N such that f , g k [ x 1 , , x M ], so if we think of f g = 0 in k [ x 1 , , x M ], we get that f = 0 or g = 0, and we are done.
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Answers (1)

grubijanebb
Answered 2022-07-11 Author has 10 answers
For instance, k [ X n : n N ] / ( X n 2 : n N ): the ring of polynomials in infinitely many variables quotiented by the ideal generated by the square of the variables.
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