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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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