Exercise 111.22.4. Show that the class groups of the following rings are trivial

a polynomial ring $A = k[x]$ where $k$ is a field,

the integers $A = \mathbf{Z}$,

a polynomial ring $A = k[x, y]$ where $k$ is a field, and

the quotient $k[x, y]/(xy)$ where $k$ is a field.

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