# Existence of module of finite injective dimension

At p. 107 of the book Cohen-Macaulay Rings by Bruns and Herzog, the authors write

"any module of finite projective dimension (over a Gorenstein ring $R$) has finite injective dimension as well, simply because $R$ has finite injective dimension by definition."

Could someone please explain why this conclusion is true?

Because a Noetherian ring $R$ has finite injective dimension if and only all R-modules with finite projective dimension also have finite injective dimension. See the proof of this on page $8$, and also the Theorem below there in http://www.math.hawaii.edu/~lee/homolog/Goren.pdf.