Homework 2 corrections

Question 2.

One can define exterior powers over commutative rings + good properties. So if:

$P\oplus R^{n-1}\simeq R^{n}\Rightarrow P\simeq P\otimes _{R}R\simeq \wedge _{R}^{1}P\otimes _{R}\wedge _{R}^{n-1}R^{n-1}\simeq \wedge _{R}^{n}R^{n}\simeq R~$

since rank of $P~$ is one ( so constant ) over $R~$ then $\wedge _{R}^{k}P=0\;\forall \;k>1~$. [ I have assumed in all this that $R~$ is unital. ]

For reference you may look at the K-book chapter 1, pages: 4 (ex 1.6), 15, 16

For discussion of rank look at the K-book chapter 1, pages: 2, 9

Exercise. If $R~$ has the invariant basis property (IBP), i.e. $R^{n}\simeq R^{m}\Leftrightarrow n=m~$, then the rank of a stably free $R~$-module $P~$ defined by ${\text{rank}}_{R}(P)=m-n~$ where $m,n\in \mathbb {N} \cup {0}~$ are such that $P\oplus R^{n}\simeq R^{m}~$ is well defined.