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If R1, R2, . . . , Rn are commutative rings with unity,

Chapter 12, Problem 24E

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QUESTION:

If \(R_1, R_2, \ldots , R_n\) are commutative rings with unity, show that \(U\left(R_{1} \oplus R_{2} \oplus \cdots \oplus R_{n}\right)=U\left(R_{1}\right) \oplus U\left(R_{2}\right) \oplus \cdots \oplus U\left(R_{n}\right)\).

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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

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