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Izumi’s theorem. (English) Zbl 0741.13011
Commutative algebra, Proc. Microprogram, Berkeley/CA (USA) 1989, Publ., Math. Sci. Res. Inst. 15, 407-416 (1989).
[For the entire collection see Zbl 0721.00007.]
The author gives a generalized version of Izumi’s theorem, which solved the old problem of characterizing analytically irreducible local domains. Let $$(Q,{\mathfrak m})$$ be a local ring and $$I$$ an $${\mathfrak m}$$-primary ideal. Then an integer-valued function $$I(x)$$ on $$Q$$ is defined by: $$I(x)=n$$ if $$x\in I^ n\backslash I^{n+1}$$ and $$I(x)=\infty$$ if $$x=0$$. Using the notation above, the author proves the following theorem, originally shown by Shuzo Izumi for analytic algebras:
Izumi’s theorem. $$Q$$ is analytically irreducible if for at least one $${\mathfrak m}$$-primary ideal $$I$$, and only if, for all $$I$$, there exists constants $$C(I)$$, $$C'(I)$$ depending only on $$I$$ such that $$I(xy)-I(y)\leq C(I)I(x)+C'(I)$$, for all $$x,y\neq 0$$ in $$Q$$.

##### MSC:
 13Hxx Local rings and semilocal rings 13J07 Analytical algebras and rings 13H99 Local rings and semilocal rings 13J10 Complete rings, completion 13B35 Completion of commutative rings