What is the difference between $\mathbb{Z}_n$ and $\mathbb{Z}/n\mathbb{Z}$ (as groups, their set structures and algebraic structures)?

Asked Apr 09 '20 at 20:49

Active May 01 '20 at 20:26

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Can you please explain this with simple examples?

To me, their elements are same. Because if I divide $ k+m\Bbb{Z}$ by $ n$, then it gives any one of the elements of $ \Bbb{Z}_n$. So can I say $\mathbb{Z}_n$ and $\mathbb{Z}/n\mathbb{Z}$ both are same? Or does it happen due to the fact that $\mathbb{Z}_n$ and $\mathbb{Z}/n\mathbb{Z}$ are isomorphic?

Also are their set structure same? (if not, then I suppose they are not same groups/rings.)

Thanks in advance.

edited May 01 '20 at 20:26

asked Apr 09 '20 at 20:49

Manjoy Das

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Does this answer your question? Difference between $\mathbb{Z}/n\mathbb{Z}$ and $\mathbb{Z}_n$ – For the love of maths Apr 09 '20 at 20:51
As far as I know, there's usually none. However $\Bbb Z_n$ may also denote the ring of fractions with denominator a power of $n$ (e.g. $\Bbb Z_{10}$ might denote the ring of decimal numbers). – Bernard Apr 09 '20 at 20:54
Are you asking about groups or rings? What is your definition of $\Bbb Z_n$? Your question might be reopened if your reply indicates it differs from the linked dupe. – Bill Dubuque Apr 09 '20 at 22:24
@BillDubuque i asked about rings. yes this question was answered before but i didn't understand their logics. – Manjoy Das Apr 10 '20 at 12:01

What is the difference between $\mathbb{Z}_n$ and $\mathbb{Z}/n\mathbb{Z}$ (as groups, their set structures and algebraic structures)?

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