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We prove that a finite dimensional algebra $A$ with representation-finitesubcategory consisting of modules that are semi-Gorenstein-projective and$n$-th syzygy modules is left weakly Gorenstein. This generalises a theorem ofRingel and Zhang who proved the result in the case $n=1$. As an application weshow that monomial algebras and endomorphism rings of modules overrepresentation-finite algebras are weakly Gorenstein. We then give a newconnection between the theory of dominant dimension and Gorenstein homologicalalgebra for gendo-symmetric algebras. As an application, we will see that theexistence of a non-projective Gorenstein-projective-injective module in agendo-symmetric algebra already implies that this algebra is not CM-finite. Weapply out methods to give a first systematic construction of non-weaklyGorenstein algebras using the theory of gendo-symmetric algebras. Inparticular, we can construct non-weakly Gorenstein algebras with an arbitrarynumber of simple modules from certain quantum exterior algebras such as theLiu-Schulz algebra.

Gendo-symmetric algebras, dominant dimensions and Gorenstein homological algebra