Local rings with bounded ideals (Q795108)

The author proves that for a local Noetherian ring (R,\({\mathfrak m})\) and a positive integer n the following three statements are equivalent: (i) Each ideal of R can be generated by n elements. (ii) If I is a proper ideal of R, then either I can be generated by n-1 elements or I is the sum of a power of m and an ideal which can be generated by n-1 elements. (iii) Each ideal of R/\({\mathfrak m}^{n+1}\) can be generated by n elements. If R is a one-dimensional Cohen-Macaulay ring, then each of these conditions is equivalent to (iv) \({\mathfrak m}^ n\) can be generated by n elements. As a special case one gets the result that each ideal of R can be generated by two elements if and only if \({\mathfrak m}\) and \({\mathfrak m}^ 2\) can each be so generated. There is no analogous result for \(n=3.\) Some hints at the noncommutative situation are given.