Convenient Notations for Free Abelian Group Is there a convenient (and commonly used) notation for Free Abelian Group? For instance, the free abelian group generated by two elements: $\langle a,b\mid ab=ba\rangle$, it seems quite unwieldy to keep writing this. Also, I wish to keep track of the generators, $a,b$, hence something like $F_2$ is not suitable. I have seen something like $\mathbb{Z}\\{a,b\\}$, but not sure if it standard? Are there any other possibilities? I am quite new to this subject. Thanks!

Up to isomorphism, the free abelian group generated by $n$ elements is $\mathbb Z^n$. If you want to keep track of generators, some standard notations are $$ \mathbb Za_1+\cdots +\mathbb Za_n $$ and $$ \langle a_1,\ldots,a_n\rangle $$ and $$ \mathbb Za_1\oplus \cdots\oplus \mathbb Z a_n, $$ where the group has been written as a direct sum of its infinite cyclic subgroups $\mathbb Z a_1=\langle a_1\rangle$ up through $\mathbb Z a_n=\langle a_n\rangle$.

Since abelian groups are the same thing as $\mathbb Z$-modules, many of these notations are special cases of notation used more generally for direct sums and direct products of $R$-modules, where $R$ is a ring.