Here is the thm.
I want to prove it and I got a hint to use the universal coefficient theorem. I am confused about which statement of a universal coefficient theorem should I use and how. Here are the statements that I know on Wikipedia https://en.wikipedia.org/wiki/Universal_coefficient_theorem for homology and cohomology. Also, I know this statement of the universal coefficient theorem from Harpreet Bedi lecture 8 called " homology to cohomology " in the homology series on youtube in this link https://www.youtube.com/watch?v=mvf8Pg26JLA&list=PL7BFF10190F42006E&index=8 : $$H^{p} (K; \mathbb{Z}) \cong Hom (H_{p}(K), \mathbb{Z}) \oplus Ext (H_{p-1}(K, \mathbb{Z}))$$
I am guessing that the statement of Harpreet Bedi is the one that should be used but I do not know how this statement comes from the one on Wikipedia and how to use it to prove my theorem. Could anyone help me with this, please?
The statement from wikipedia is more precise, but here either statement is good enough for the result.
You just need to know what $\mathrm{Ext}^1_R($a free module,$R)$ is. This should be covered in any lecture about the universal coefficients theorem.