semiprime ideals in rings without identity

Question

I have found the following proposition in Lam's Noncommutative Algebra. The proof was declared as "trivial", but i could not figure it out for myself for rings without $1$.

Let $R$ be a ring and $I$ a ideal of $R$. The following statements are equivalent.

(1) For $a \in R$, $(a)^2\subseteq I$ implies $a\in I$.

(2) For $a \in R$, $aRa\subseteq I$ implies $a \in I$.

(I only have trouble with (1) implies (2).)

Answer 1

The proof was declared as "trivial", but i could not figure it out for myself for rings without 1

Lam's book does not claim it holds for rings without identity: if he discusses rings without identity at all, it is confined to isolated parts of the book. Certainly it wasn't intended to develop a theory of prime and semiprime ideals in rings without identity.