Abstract: We introduce two kinds of new variations of dominant dimensions, which have some advantages in the study of gendo-Gorenstein algebras. One is the $\nu$-stably dominant dimension associated to the Nakayama functor $\nu$. Using it, a criterion for an algebra being gendo-Gorenstein is given, and the gendo-Gorensteiness of algebras are invariant under left-split extensions. Moreover, the difference of the $\nu$-stably dominant dimension is bounded for two derived equivalent gendo-Gorentein algebras. The other is a dominant dimension building from the Gorenstein balance module. An upper bound of this kind of dominant dimension is given for gendo-Gorenstein algebras.

Key words: Nakayama functor, stably dominant dimension, gendo-Gorenstein algebra