(a) If A is a module over a commutative ring Rand a A, is an ideal...

(a) If A is a module over a commutative ring Rand a ϵ A,  is an ideal of R. If a is said to be a torsion element of A.

(b) If R is an integral domain, then the set T(A) of all torsion elements of A is a submodule of A. (T(A) is called the torsion submodule.)

(c) Show that (b) may be false for a commutative ring R, which is not an integral domain. In (d) - (f) R is an integral domain.

(d) If ∫ : A → B is an R-module homomorphism, then ∫(T(A)) ⊂ T(B); hence the restriction ∫_T of f to T(A) is an R-module homomorphism T(A) → T(B). 

(e) If is an exact sequence of R-modules, then so is (f) If g : B → C is an R-module epimorphism, then g_T : T(B) → T( C) need not be an epimorphism. 

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