To prove that the divisors of 1 are units in the ring $R$, we need to understand what it means for an element to be a unit in a ring and what it means for an element to be a divisor of 1 in the ring $R$.

Definitions

1. Unit: An element $u$ in a ring $R$ is called a unit if there exists an element $v$ in $R$ such that $uv = 1$. The element $v$ is called the multiplicative inverse of $u$, and $u$ is invertible.
2. Divisor of 1: An element $d$ in a ring $R$ is called a divisor of 1 if there exists an element $e$ in $R$ such that $de = 1$.

Proof

1. Suppose $d$ is a divisor of 1 in the ring $R$. By definition, there exists an element $e$ in $R$ such that $de = 1$.

2. The equation $de = 1$ indicates that $d$ has a multiplicative inverse $e$ in the ring $R$.

3. According to the definition of a unit, $d$ is a unit if there exists some element $e$ in $R$ such that $de = 1$. Since we have $de = 1$ by the assumption that $d$ is a divisor of 1, it follows that $d$ is a unit.

4. Therefore, any element $d$ that is a divisor of 1 in $R$ must be a unit in $R$, because it satisfies the definition of a unit by having a multiplicative inverse $e$ such that $de = 1$.

Conclusion

We have shown that if an element $d$ in a ring $R$ is a divisor of 1, then $d$ must be a unit in $R$. This completes the proof.

Would you like any further details or have any questions?

Here are some questions you might be interested in asking:

1. What are some examples of rings and their units?
2. Can every element in a ring be a unit?
3. What is the difference between a unit and a non-unit element in a ring?
4. How does the concept of units apply to fields?
5. Can you explain more about the structure of rings and their elements?

Tip: When dealing with ring theory, always pay close attention to the definitions and properties of elements like units, zero divisors, and ideals. This precision helps in understanding more complex concepts and proving theorems effectively.