Let $\Lambda$ be a finite dimensional $k$-algebra over an algebraically closed field $k$. $T$ is a tilting module, then for an $\Lambda$-module $X$, I have seen in a material that
"consider the universal extension $$0 \rightarrow X \rightarrow E \rightarrow T' \rightarrow 0$$ with $T' \in add T$. This extension has the property that the connecting homomorphism $Hom_{\Lambda}(T,T') \rightarrow Ext_{\Lambda}^1(T,X)$ is surjective."
I have not seen universal extension before. Could anyone provide me some materials about universal extensions containing the above property?
The best detailed reference can probably be found in the proof of lemma 2.4. in chapter VIII. in the book Frobenius algebras 2 by Skowronski and Yamagata. It is not called universal extension there but it is explained very well how it is constructed.