The definition of Jacobson radical

Question

I see 2 definition of Jacobson radical in A First Course in Noncommutative Algebra of T.Y.Lam but I wonder if it is the same.

Give $I$ is an ideal in $R$ called modular if there exist $e\in R$ such that $\forall r\in R$, $re-r\in I$. When ${M}$ is left maximal modular of $R$, we have $\cap(M)= rad (R)$. This is the first definition and the second one is $rad(R)=\cap N$ with N is left maximal ideal of $R$.

So is it the same? I am trying to prove every element in $rad(R)=\cap N$ with N is left maximal ideal of $R$ is quasi-regular. It means $x+y=-xy=-yx$ for all $x,y\in R$.

Answer 1

In a ring with identity (Lam's context) every left ideal is modular since $e=1$ works.

The modularity condition is necessary when working with rings without identity (Jacobson's context) to rule out bad cases, e.g. $2\mathbb Z/4\mathbb Z$, in which the zero ideal is a maximal but nonmodular ideal.

So Jacobson's version is more general.