  We prove that a finite dimensional algebra $A$ with representation-finite
subcategory consisting of modules that are semi-Gorenstein-projective and
$n$-th syzygy modules is left weakly Gorenstein. This generalises a theorem of
Ringel and Zhang who proved the result in the case $n=1$. As an application we
show that monomial algebras and endomorphism rings of modules over
representation-finite algebras are weakly Gorenstein. We then give a new
connection between the theory of dominant dimension and Gorenstein homological
algebra for gendo-symmetric algebras. As an application, we will see that the
existence of a non-projective Gorenstein-projective-injective module in a
gendo-symmetric algebra already implies that this algebra is not CM-finite. We
apply out methods to give a first systematic construction of non-weakly
Gorenstein algebras using the theory of gendo-symmetric algebras. In
particular, we can construct non-weakly Gorenstein algebras with an arbitrary
number of simple modules from certain quantum exterior algebras such as the
Liu-Schulz algebra.
