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We shall bring two possible criteria, each shows when $R=S^2$. The first criterion: There exist $x,y \\in S$ such that $xy-yx \\neq 0$ and $xSy \\subseteq S^2$ $\\Leftrightarrow$ $S^2=R$. The second criterion: There exist $x,y \\in S$ such that $xy+yx \\neq 0$ and $xKy \\subseteq S^2$ $\\Leftrightarrow$ $S^2=R$. 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We shall see that in such a ring $R$, $R=S^3$. We shall bring two possible criteria, each shows when $R=S^2$. The first criterion: There exist $x,y \\in S$ such that $xy-yx \\neq 0$ and $xSy \\subseteq S^2$ $\\Leftrightarrow$ $S^2=R$. The second criterion: There exist $x,y \\in S$ such that $xy+yx \\neq 0$ and $xKy \\subseteq S^2$ $\\Leftrightarrow$ $S^2=R$. 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