On Nilpotent Rings of Order $p^4$ with Some Additional Properties

In this paper, indecomposable nilpotent rings of order $p^4$ and of index $3$ satisfied the identity $x^2=0$ have been described. Moreover, indecomposable nilpotent rings of order $p^4$ and of index $3$ that have an element of additive order $p^t$ ($t\in\{3,4\}$) have been classified.