In the first part, we study algebras A such that A = R ⨿ I, where R is a subalgebra and I a two-sided nilpotent ideal. Under certain conditions on I, we show that A is standardly stratified if and only if R is standardly stratified. Next, for $A=\left[\begin{array}{cc}U& 0\\ M& V\end{array}\right]$, we show that A is standardly stratified if and only if the algebra R = U × V is standardly stratified and ${}_{V}M$ is a good V-module.

Let $A$ be a standard Koszul standardly stratified algebra and $X$ an $A$-module. The paper investigates conditions which imply that the module ${\mathrm{Ext}}_A^{*}\left(X\right)$ over the Yoneda extension algebra $A^{*}$ is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of $A$ is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

Let Λ be an artinian ring and let 𝔯 denote its Jacobson radical. We show that a simple module of finite projective dimension has no self-extensions when Λ is graded by its radical, with at most two simple modules and 𝔯⁴ = 0, in particular, when Λ is a finite-dimensional algebra over an algebraically closed field with at most two simple modules and 𝔯³ = 0.