a) If R has an identity and A is an R module, then there are submodules Band C of A such that B is unitary, RC 0 and A B C (b) Let A 1 be another R module, with A 1 B 1 C1 (B1 unitary, RC 1 0) If A A 1 is an R module homomorphism then (B) B 1 and f(C) C 1 (c) If the map of part (b) is an epimorphi...

a Let B be the submodule generated by 1A ie B r 1A r R Since R has an identity B is a unitary submod ...

{a) If R has an identity and A is an R-module, then there are submodules Band C

Question:

{a) If R has an identity and A is an R-module, then there are submodules Band C of A such that B is unitary, RC = 0 and A = B ⊕ C.

(b) Let A₁ be another R-module, with A₁ = B₁⊕C1 (B1 unitary, RC₁ = 0). If ∫: A → A₁ is an R-module homomorphism then ∫(B) ⊂ B₁ and f(C) ⊂ C₁. (c) If the map ∫ of part (b) is an epimorphism [resp. isomorphism], then so are ∫| B : B →B₁ and ∫| C : C →C₁.