Length functions in commutative rings with zero divisors

摘要

Let R be a commutative ring. We investigate several functions which measure the length of factorizations of an element of R. Some of these unctions are I,I-U : R -> N-0 (for R atomic) and L,L-U : R -> N-0 boolean OR {infinity} here I(x) = I-U(x) = L(x) = L-U(x) = 0 for x a unit, and for x not a unit I(x)= inf{n vertical bar x = x(1) ... x(n),x(i) irreducible}, I-U(x) = inf{n vertical bar x = y(1) ... y(m) x(1) ... x(n) a U-decomposition}, L(x) = sup{n vertical bar x = x(1) ... x(n),x(i) a nonunit}, and L-U(x) = sup{n vertical bar x = y(1) ... y(m)x(1) ... x(n) a U-factorization}. For R satisfying the ascending chain condition on principal ideals, we consider the ordinal-valued function (L) over tilde obtained by recursively defining (L) over tilde (x) to be the least ordinal strictly greater than (L) over tilde (y) for each proper divisor y of x.