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We give some sufficient and necessary conditions for an element in a ring to be an EP element, partial isometry, normal EP element and strongly EP element by using solutions of certain equations.

Let $r\left({𝔴}_2^0\right)$ be the Green ring of the weak Hopf algebra ${𝔴}_2^0$ corresponding to Sweedler’s 4-dimensional Hopf algebra $H_2$, and let $\mathrm{Aut}\left(R\left({𝔴}_2^0\right)\right)$ be the automorphism group of the Green algebra $R\left({𝔴}_2^0\right)=r\left({𝔴}_2^0\right){\otimes }_{\mathbb{Z}}ℂ$. We show that the quotient group $\mathrm{Aut}\left(R\left({𝔴}_2^0\right)\right)/C_2\cong S_3$, where $C_2$ contains the identity map and is isomorphic to the infinite group $({ℂ}^{*},\times )$ and $S_3$ is the symmetric group of order 6.