If M=N+mR then use zorn's lemma to show that N is contained in a maximal submodule k of M.where N is submodule of M

Question

It is about modules..

Answer 1

 

 

 

Traditional , old school set theory.

 

let x be an element of N and
n = sup(N), by Zorn's Lemma, which states there
must be a greatest element or upper bound.

Then x<=n because X <=n for all X in N.

Then x is also an element of M=N + mR, as
M = N U mR. So x must be in the union.

So if x is in N, then x is in M.
Hence N is a subset of M.

But n = sup(N) which is also in M
since N is a subset of M.

Since x<=n contained in N contained in M,
x is inside a subset of M.