Can every abelian group converted into ring(by defining multiplication operation) with identity with same order. We can convert every group G into ring by defining a.b = 0 for all a and b in G. But this ring has no multiplicative identity
Asked
Active
Viewed 283 times
1 Answers
1
The classification of finitely generated abelian groups implies that the group is a direct sum of copies of $\mathbb{Z}$ and copies of $\mathbb{Z}/n\mathbb{Z}$ for certain natural numbers $n \gt 1$. Endowing these groups with the natural ring structure coordinatewise gives a nontrivial ring structure.
Nicky Hekster
 42,900
 7
 54
 93

What about the non finitely generated case ? – Amr Oct 17 '21 at 19:41