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## Main Question or Discussion Point

Notations:

M denotes an abelian group under addtion

R denotes a commutative ring with identity

ann(?): Let M be an R-module and r∈R v∈M, then ann(v)={r∈R | rv=0}

Terms:

R-module: a module whose base ring is R

torsion element: A nonzero element v∈M for which rv=0 for some nonzero r∈R

torsion module: all elements of the module are torsion elements

integral domain: a commutative ring R with identity with the property that for r,s∈R, rs≠0 if r≠0 and s≠0

Question:

Find a torsion module M for which ann(M)={0}.

I wonder whether the base ring R of such M can only be an integral domain.

M denotes an abelian group under addtion

R denotes a commutative ring with identity

ann(?): Let M be an R-module and r∈R v∈M, then ann(v)={r∈R | rv=0}

Terms:

R-module: a module whose base ring is R

torsion element: A nonzero element v∈M for which rv=0 for some nonzero r∈R

torsion module: all elements of the module are torsion elements

integral domain: a commutative ring R with identity with the property that for r,s∈R, rs≠0 if r≠0 and s≠0

Question:

Find a torsion module M for which ann(M)={0}.

I wonder whether the base ring R of such M can only be an integral domain.