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An integral domain $D$ is called a Euclidean domain if there is a function $d$ from the non-zero elements of D to the non-zero integers such that

1) $d\left ( a \right ) \leq d\left ( ab \right ) \forall a,b \in D$.

2) if $a,b \in D, b\neq 0$, then there exists elements $q$ and $r$ in $D$ such that $a=bq+r$, where $r=0$ or $d\left ( r \right )<d\left ( b \right )$.

In the second condition, it is mentioned that $b$ is non-zero. This is tautological. It follows directly from the map $d$ that $d$ maps only non-zero elements. Or is there something subtle to this?

Xam
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Mathematicing
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    It is not tautological. If you postulate "If $a,b\in D$, then there exist elements $q$ and $r$ in $D$ such that $a=bq+r$, where $r=0$ or $d(r)<d(b)$", then in particular you postulate that there exist $q,r\in D$ such that $1 = 0\cdot q+r$ where $r=0$ or $d(r)<d(0)$, so as $d(0)$ is not even defined it seems you postulate that $1=0\cdot q+0$. – Hagen von Eitzen Jul 16 '17 at 12:58

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There isn't really subtle behind the requirement that $b\neq 0$. Algebraic structures are nice generalizations of known structures. So when defining an abstract structure we want that structure behaves nicely as it does in the known structure. In the case of Euclidean domains, we define them in order to mimic the division algorithm for the integers. Now, as you surely know, in $\Bbb Z$ we can't divide by zero, so going to the abstract concept of Euclidean domains, we demand that the "divisor", i.e., $b$ must be nonzero like it happens in the integer case. So, this is basically the reason why we take $b\neq 0$.

On the other hand, the definition of Euclidean domain you have is maybe the standard one, but is not the only definition that exists. According to other authors we can define $D$ to be a Euclidean domain if there is a map $d\colon D\rightarrow \Bbb N$ (here $\Bbb N=\{0,1,\ldots\}$) that satisfies:

i) $d(0)=0$ and $d(a)>0$ for all $a\neq 0$.

ii) For $a,b\in D$, $b\neq 0$, there are $q,r\in D$ such that $a=bq+r$ and $d(r)<d(b)$.

This definition can be found for example in Barshay's book "Topics in Ring Theory". Actually, the definition of Euclidean domains have changed along the time, see for example this paper where the authors discuss different definitions (and generalizations) of Euclidean domains.

Finally, the item 1) given in your definition is sometimes called the "sub-multiplicative property" of the euclidean function. In this blurb K. Conrad shows that 1) isn't esencial in the definition of Euclidean domain. This is also discussed in the Wikipedia page of Euclidean domains.

Xam
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