MHB Corollary to Correspondence Theorem for Modules

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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
 
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Hint: when $R$ itself is considered as a (left) $R$-module, the submodules of $R$ are precisely the (left) ideals of $R$.
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
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