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Anonymous

Subject : abstract Algebra

HW question :Prove that if M is Maximal, then R/M is simple.

*I have a guidline below (hint) how to solve it.

----------------------------------

Suppose that M is a maximal ideal in R.

Also suppose that J is an ideal in R/M.

Let I = {a in R : a+M is in J}.

Prove that I is an ideal in R.

Prove that M is a subset of I, and I is a subset of R.

Since M is maximal, this implies that either I=M or I=R.

Use that to prove that either J = {M} or J = R/M.

Hence the only ideals in R/M are the trivial ones (since {M}

is the ideal in R/M that contains the zero element of R/M).

I put the homemwork question. its just one sentence : Prove that if M is Maximal, then R/M is simple.

I also put the Hint how to prove, so basically thats the answer, but have to put detail of the proof

I attached the file if expert need more Hint

The prove should be done follwing the guideline I wrote? Because before, I tried other ways to prove(using Field) but prof said its wrong.

■Question :
-----------------------------------------------------------------------------------------------------------------------------------------■How to prove : guideline.
Suppose that M is a maximal ideal in R.
Also suppose that J is an ideal in R/M.
Let I = {a in R : a+M is in J}.
Prove that I is an ideal in R.
Prove that M is a subset of I, and I is a subset of R.
Since M is maximal, this implies that either I=M or I=R.
Use that to prove that either J = {M} or J = R/M.
Hence the only ideals in R/M are the trivial ones (since {M}
is the ideal in R/M that contains the zero element of R/M).
■Teacher’s comment(Hint)
You can't say that R/M is a field and then argue from there. That result is a corollary of this
problem, so you can't use the corollary to prove the theorem. Instead, you need to argue
directly. The other direction of the theorem is proved in the lecture notes for Section 4.4, and
there are hints in the lecture notes for how to do this problem.
Below is the lecture notes.
■Teacher’s comment :
The hypotheses to this problem only say that R is a ring. We're not assuming that R is
commutative, and we're not assuming that it contains a multiplicative identity. So there doesn't
have to be any element 1 in R. And therefore you're not going to be able to prove that R/M is a
field ...
Instead, use the hints from the lecture notes:
Suppose that M is a maximal ideal in R.
Also suppose that J is an ideal in R/M.
Let I = {a in R : a+M is in J}.
Prove that I is an ideal in R.
Prove that M is a subset of I, and I is a subset of R.
Since M is maximal, this implies that either I=M or I=R.
Use that to prove that either J = {M} or J = R/M.
Hence the only ideals in R/M are the trivial ones (since {M}
is the ideal in R/M that contains the zero element of R/M).
■ Definition of Simple Ring, Maximal, Ideal
...

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