Let $I$ be a semiprime ideal of $\mathbf{A}$. Suppose $a\mathbf{A}a \subseteq I$ for some $a$. Show that $a \in I$.

Question

Let $\mathbf{A}$ be an algebra. An ideal $I$ of $\mathbf{A}$ is called semiprime if $\mathbf{A}/I$ has no nonzero nilpotent ideal. Can someone supply a proof of following fact?

Let $I$ be a semiprime ideal of $\mathbf{A}$. Suppose $a\mathbf{A}a \subseteq I$ for some $a$. Show that $a \in I$.