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## Homework Statement

I am curious,

if I,J, and M are ideals of the commutative ring R, and M/I[itex]\subseteq[/itex]J/I, then M[itex]\subseteq[/itex]J

## Homework Equations

M/I = { m+I : m is in M}

J/I = { j+I : j is in J}

I[itex]\subseteq[/itex]R is an ideal if

1.) if a and b are in I then a+b is in I

2.) if r is in R and a is in I then a*r is in I

## The Attempt at a Solution

Proof by contradiction:

Assume M/I[itex]\subseteq[/itex]J/I,

and M is not contained in J.

Since M is not contained in J then there exists a m in M that is not in J.

Then m+I is in M/I but not in J/I.

This is a contradiction to the hypothesis that M/I[itex]\subseteq[/itex]J/I. Thus M is contained in J. (QED)

Any flaws?

Thank you for your time.

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