Is the quotient by a unit in an euclidean domain unique?

Asked Mar 30 '23 at 02:34

Active Mar 30 '23 at 15:35

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I've seen some conditions for quotients and reminders to be unique, though they are not in general. However I haven't been able to find a counter example for the case when "dividing" by a unit. In this case, is the quotient and reminder unique? ($a=a\times u+0$, where $u$ is a unit).

For my defintion of euclidean domain I'm using Aluffi's:

edited Mar 30 '23 at 15:35

asked Mar 30 '23 at 02:34

Fernando Chu

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I think it must be unique if $1$ is defined to be the multiplicative identity and $0$ is the additive identity. – person Mar 30 '23 at 02:42
There are many different definitions of Euclidean domains. Please supply the precise definition you are using – Bill Dubuque Mar 30 '23 at 02:51
What you might want to ask is "is the quotient by a unit in a euclidean domain unique?" 1 is an ambiguous word. Rings do not concern themselves with multiplicative identities, but rather with units, which are more general. – Fomalhaut Mar 30 '23 at 03:00
thank you all, I've edited the question accordingly – Fernando Chu Mar 30 '23 at 03:06
What is the definition of "valuation" being used in that definition? (and what is footnote $8$?) – Bill Dubuque Mar 30 '23 at 03:14
@BillDubuque A valuation is just a function from $R \setminus 0$ to the non-negative integers. Footnote 8 just points out that $v(b)\leq v(ab)$ is sometimes required but not really needed. – Fernando Chu Mar 30 '23 at 14:10

Is the quotient by a unit in an euclidean domain unique?

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