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Modules, Submodules, Commutative Rings with Unity and Homomorphism

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1. Let R and S be commutative rings with unity, and let φ: R S be a ring homomorphism. If M is an S-module, prove that M is also an R-module if we define rm = φ(r)m for all r E R and m E M.

2. If M1 and M2 are submodules of an R-module M such that M = M1(+)M2, prove that M1 = M/M2 and M2 = M/M1.

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Modules, Submodules , Commutative Rings with Unity and Homomorphism are investigated.

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1. Proof:
To show that is an -module, we have to verify the following conditions.
(a) For any , , .
Since is an -module, then we have

(b) For any , ,
Since is a ...

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