Subrings in a Ring with Positive Characteristic

If a ring has a positive characteristic $n$, show that it contains a subring isomorphic to $\mathbb{Z}_n$.

In this problem, we explore the concept of rings with positive characteristics and the existence of subrings within them. A ring with a positive characteristic means that there is a smallest positive integer n such that n times the multiplicative identity, which is assumed to be 1 in the ring, equals the additive identity, or zero, in the ring. This property is crucial for understanding how the arithmetic within the ring behaves in comparison to integer arithmetic, particularly under addition. The concept of characteristic is analogous to considering arithmetic modulo n, where numbers wrap around upon reaching a certain magnitude, similar to how clock arithmetic works. In rings with positive characteristic n, this wrap-around behavior implies a natural resemblance to modular arithmetic, and this is precisely why such a ring contains a subring isomorphic to the integers modulo n, denoted as $Z_n$.

In exploring why a subring isomorphic to $Z_n$ exists, it’s helpful to consider the map that sends an integer k to k times the multiplicative identity in the ring. This map respects both addition and multiplication, making it a homomorphism from integers to the ring. By focusing on the kernel of this map, which is characterized by the nature of the ring’s characteristic, we can identify a subring that behaves identically to $Z_n$. This property illustrates one of the beautiful interplays between algebraic structures and reveals fundamental insights into the nature of rings.