Corollary to Correspondence Theorem for Modules

[Math Amateur]

I am reading Joseph J. Rotman's book: Advanced Modern Algebra and I am currently focused on Section 6.1 Modules ...

I need some help with the proof of Corollary 6.25 ... Corollary to Theorem 6.22 (Correspondence Theorem) ... ...

Corollary 6.25 and its proof read as follows:View attachment 4925Can someone explain to me exactly how Corollary 6.25 follows from the Correspondence Theorem for Modules ...?

Hope that someone can help ...

Peter=============================================

*** EDIT ***

The above post refers to the Correspondence Theorem for Modules (Theorem 6.22 in Rotman's Advanced Modern Algebra) ... so I am proving the text of the Theorem from Rotman's Advanced Modern Algebra as follows:View attachment 4926

 

[Deveno]

Hint: when $R$ itself is considered as a (left) $R$-module, the submodules of $R$ are precisely the (left) ideals of $R$.