Solution Found!
If R1, R2, . . . , Rn are commutative rings with unity,
Chapter 12, Problem 24E(choose chapter or problem)
QUESTION:
If \(R_1, R_2, \ldots , R_n\) are commutative rings with unity, show that \(U\left(R_{1} \oplus R_{2} \oplus \cdots \oplus R_{n}\right)=U\left(R_{1}\right) \oplus U\left(R_{2}\right) \oplus \cdots \oplus U\left(R_{n}\right)\).
Questions & Answers
QUESTION:
If \(R_1, R_2, \ldots , R_n\) are commutative rings with unity, show that \(U\left(R_{1} \oplus R_{2} \oplus \cdots \oplus R_{n}\right)=U\left(R_{1}\right) \oplus U\left(R_{2}\right) \oplus \cdots \oplus U\left(R_{n}\right)\).
ANSWER:Step 1 of 2
Let first denote the unity element in each rings , i.e., is unity element in since if .
This means there is element such that