Homomorphism from Integers to Ring with Unity

Prove that the mapping from the integers $\mathbb{Z}$ into a ring with unity, where an integer $n$ maps to $n \times 1$, preserves addition and multiplication, and therefore is a homomorphism.

To prove that a mapping is a homomorphism, we need to ensure it preserves the structure of the algebraic systems involved. In this case, we start with the integers and a ring with unity. The mapping described takes an integer and maps it to the product of that integer with the unity element of the ring. This specific setup touches on the concept of ring homomorphisms. The essence of a ring homomorphism is that it maintains the operations of addition and multiplication from the domain structure in the codomain structure.

For a mapping to be a homomorphism, it must preserve both addition and multiplication. The intuition is that operations performed on elements in the original set should yield consistent results when mapped to the images in the target set. By taking an integer n and mapping it to n times the unity element of the ring, we essentially redefine how addition and multiplication are visualized within this new structure. The operation of multiplication by unity inherently takes the integer nature of the input and anchors it within the multiplicative framework of the ring.

Thus, understanding this homomorphism requires considering how basic operations in the integers translate into the operations of the target ring. Preserving addition assures us that the homomorphism respects the linearity in the domain, while multiplication preservation guarantees the embedding of the integer structure within the ring. The mapping essentially serves as a bridge between pure number theory concepts and the more abstract ring structures encountered in higher algebra.