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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