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Take $A$ to be a regular quadratic algebra of global dimension three. Weobserve that there are examples of $A$ containing a dimension three regularcubic algebra $C$. If $B$ is another dimension three regular quadratic algebra,also containing $C$ as a subalgebra, then we can form the pushout algebra $D$of the inclusions $i_1:C\hookrightarrow A$ and $i_2:C\hookrightarrow B$. Weshow that for a certain class of regular algebras $C\hookrightarrow A,B$, theirpushouts $D$ are regular quadratic algebras of global dimension four.Furthermore, some of the point module structures of the dimension threealgebras get passed on to the pushout algebra $D$.

Artin-Schelter Regular Algebras, Subalgebras, and Pushouts