is a left -module homomorphism from to . In other words, the mapping is an element of . We call this mapping , since it only depends on . For any , the mapping
is a then a right -module homomorphism from to . Let us call it .
Definition. Let , , and be given as above. If is injective, we say that is torsionless. If is in addition an isomorphism, we say that is reflexive. A torsionless module is sometimes referred to as being semi-reflexive.
Some of the properties of torsionless and reflexive modules are
any free module is torsionless.
based on the two properties above, any projective module is torsionless.
any finite direct sum of reflexive modules is reflexive; any direct summand of a reflexive module is reflexive.
based on the two immediately preceding properties, any finitely generated projective module is reflexive.
|Date of creation||2013-03-22 19:22:38|
|Last modified on||2013-03-22 19:22:38|
|Last modified by||CWoo (3771)|