To complement Mariano's answer: If finite projective dimension implies finite injective dimension for any module $M$, then $R$ better has finite injective dimension (the converse is also quite easy).

The local rings $R$ which have finite inj. dim. over themselves are also known as Gorenstein rings. In fact, a theorem by Foxby says that $R$ possesses a module of both finite proj. and inj. dim. if and only if $R$ is Gorenstein.

To complement Mariano's answer: If finite projective dimension implies finite injective dimension for any module $M$, then $R$ better have finite injective dimension (the converse is also quite easy).

The local rings $R$ which have finite inj. dim. over themselves are also known as Gorenstein rings. In fact, a theorem by Foxby says that $R$ possesses a module of both finite proj. and inj. dim. if and only if $R$ is Gorenstein.

To complement Mariano's answer: If finite projective dimension implies finite injective dimension for any module $M$, then $R$ better has finite injective dimension (the converse is also quite easy).

The local rings $R$ which have finite inj. dim. over themselves are also known as Gorenstein rings. In fact, a theorem by Foxby says that $R$ possesses a module of both finite proj. and inj. dim. if and only if $R$ is Gorenstein.

To complement Mariano's answer: If finite projective dimension implies finite injective dimension for any module $M$, then $R$ better has finite injective dimension (the converse is also quite easy).

The local rings $R$ which have finite inj. dim. over themselves are also known as Gorenstein rings. In fact, a theorem by Foxby says that $R$ possesses a module of both finite proj. and inj. dim. if and only if $R$ is Gorenstein.