**Hopefully someone can clarify the following:**

Regarding the conditions $ \displaystyle J(R) = R$ and $ \displaystyle J(R) = 0$ ... ... ... ... I am still a little confused ...

Now, regarding $ \displaystyle \text{ Rad}(M)$ ... Bland is clear when it comes to $ \displaystyle \text{ Rad}(M) = M$ ... indeed Bland's definition of $ \displaystyle \text{ Rad}(M)$ is as follows:

"... ... If $ \displaystyle M$ is an $ \displaystyle R$-module, then the radical of $ \displaystyle M$, denoted by $ \displaystyle \text{ Rad}(M)$, is the intersection of the maximal submodules of $ \displaystyle M$. If $ \displaystyle M$ fails to have maximal submodules, then we set $ \displaystyle \text{ Rad}(M) = M$. ... ...

Presumably $ \displaystyle \text{ Rad}(M) = \{ 0 \}$ when there exist maximal submodules in $ \displaystyle M$, but their intersection is $ \displaystyle \{ 0 \}$

Now with $ \displaystyle J(R)$ Bland does not explicitly give a condition for which $ \displaystyle J(R) = R$ ... indeed his definition of the Jacobson radical is as follows:

"... ... The Jacobson radical of $ \displaystyle R$, denoted $ \displaystyle J(R)$, is the intersection of maximal right ideals of $ \displaystyle R$. If $ \displaystyle J(R) = 0$, then R is said to be a Jacobson semisimple ring ... ..."

Is it correct to assume that if there are no maximal right ideals in $ \displaystyle R$, then $ \displaystyle J(R) = R$ ... ...

... ... and $ \displaystyle J(R) = 0$ if there do exist maximum right ideals but their intersection is $ \displaystyle \{ 0 \}$

Can someone please confirm that my thinking is correct and/or point out any shortcomings or errors ...

Hope someone can help ...

Peter