What Defines the Smallest Submodule of a Module Containing a Given Subset?

[iamalexalright]

Homework Statement

Let S = \{v_{},...,v_{n}\} be a subset of a module M. Prove that
N = <<S>> = \{r_{1}v_{1} + ... + r_{n}v_{n} | r_{i} \in R, v_{i} \in S\}
is the smallest submodule of M containing S. First you will need to formulate precisely what it means to be the smallest submodule of M containing S.



don't really know where to start formulating this. A small hint would be appreciated so I can get started

 

[Robert1986]

Homework Statement

Let S = \{v_{},...,v_{n}\} be a subset of a module M. Prove that
N = <<S>> = \{r_{1}v_{1} + ... + r_{n}v_{n} | r_{i} \in R, v_{i} \in S\}
is the smallest submodule of M containing S. First you will need to formulate precisely what it means to be the smallest submodule of M containing S.



don't really know where to start formulating this. A small hint would be appreciated so I can get started

Think of something like a maximal ideal. See how it is defined and I think that you will be able to see how something is defined as the smallest submodule containing S (or subject any constraint.)

 

[iamalexalright]

I know I am looking for something similar but this is the first time I've seen maximal ideals.

So basically an ideal I in a ring R is a maximal ideal if I =/= R and if whenever J is an ideal satisfying I <= J <= R, then either J = I or J = R.

So given N = <<S>> and L <= M and N <= L and N is the smallest submodule, then L = N or L = M.