Krull Dimension is defined by induction on ordinals.

Question

I was reading the book "Serial Rings" by Gennadi Puninski, there it is written, "The Krull dimension $Kdim(M)$ of a module $M$ is defined by induction on ordinals".

I can't understand the meaning of "induction on ordinals".

I am new to all these, please help me understand the meaning of the above mentioned.

(I searched on Google but can't get the answer. If anyone can find a link please share.)

Answer 1

The full excerpt is:

The Krull dimension $\mathrm{Kdim}(M)$ of a module $M$ is defined by induction on ordinals. Set $\mathrm{Kdim}(M)=-1$ iff $M=0$. Suppose $\mathrm{Kdim}(M)=\beta$ has been already defined for every ordinal $\beta < \alpha$. Then $\mathrm{Kdim}(M)=\alpha$ if $\mathrm{Kdim}(M) \not< \alpha$ and there is no (strongly) descending chain $M_1 \supset M_2 \supset \dotsc$ of submodules [of] $M$ such that $\mathrm{Kdim}(M_i / M_{i+1}) \not< \alpha$ for every $i$. If $\mathrm{Kdim}(M) > \alpha$ for every $\alpha$, we shall say that the Krull dimension of $M$ is undefined and write $\mathrm{Kdim}(M)=\infty$.

Formally, we define the (large) set $S_{\alpha}$ of modules with Krull dimension $=\alpha$ by transfinite recursion, which is an extension of the usual recursion principle from $\mathbb{N}=\omega$ to any ordinal number. We also use the set $S_{<\alpha} := \bigcup_{\beta < \alpha} S_{\beta}$ of modules of Krull dimension $<\alpha$. So we have $M \in S_{\alpha}$ iff $M \notin S_{<\alpha}$ and there is no descending chain of submodules in $M$ whose quotients are not in $S_{<\alpha}$.