Question

Let M be a left module over a ring K. a) For each $x \in M$, the annihilator of x is defined to be the set of all $\lambda \in K$ such that $\lambda x = 0$. Show that this is a left ideal in K. b) Suppose that K is an integral domain. Show that the elements x of M whose annihilator is not {0} form a submodule T of M. (T is called the torsion submodule of M, and M is said to be torsion-free if T = {0}.) c) Show that the quotient module M/T is torsion-free. d) Calculate T when K = Z and M = Z$^2$/L, where L is the subgroup of Z$^2$ generated by the vector (4, 6).

          Let M be a left module over a ring K.
a) For each $x \in M$, the annihilator of x is defined to be the set of all $\lambda \in K$ such that $\lambda x = 0$.
Show that this is a left ideal in K.
b) Suppose that K is an integral domain. Show that the elements x of M whose annihilator
is not {0} form a submodule T of M. (T is called the torsion submodule of M, and M is
said to be torsion-free if T = {0}.)
c) Show that the quotient module M/T is torsion-free.
d) Calculate T when K = Z and M = Z$^2$/L, where L is the subgroup of Z$^2$ generated
by the vector (4, 6).

Let M be a left module over a ring K.
a) For each x ∈ M, the annihilator of x is defined to be the set of all λ∈ K such that λ x = 0.
Show that this is a left ideal in K.
b) Suppose that K is an integral domain. Show that the elements x of M whose annihilator
is not 0 form a submodule T of M. (T is called the torsion submodule of M, and M is
said to be torsion-free if T = 0.)
c) Show that the quotient module M/T is torsion-free.
d) Calculate T when K = Z and M = Z^2/L, where L is the subgroup of Z^2 generated
by the vector (4, 6).

Added by Pablo B.

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