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Just now I realized that I asked a stupid question.. Please ignore this question.

While learning Associated graded modules, I defined associated graded module, which ended in this particular situation

Say $\mathfrak I^n$ and $\mathfrak I^{n+1}$ are ideals of same ring $R$ then if I am defining quotient of these two ideals $\mathfrak I^n/ \mathfrak I^{n+1}$ should I add $\mathfrak I^{n+1} \ne (0_R)$

In other words, Let $R$ is a ring $\mathfrak {(0)}$ is Ideal generated by $0_R$, then can I say $\mathfrak {(0)}/ \mathfrak {(0)} = 0_R$

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up vote 4 down vote accepted

The quotient is going to be the zero ideal, rather than the element zero as you have written. But in answer to the main question, taking the quotient by the zero ideal is well defined (and leaves the ideal unchanged).

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Thanks for your answer. –  Ram Feb 8 '13 at 16:00

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