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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 \(Q\).

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