Elasticity properties preserved in the normset (Q700189)

The elasticity \(\rho(r)\) of an element \(r\) of a commutative semigroup \(S\) is defined as the supremum of the ratios \(n/m\), taken over all pairs of irreducible factorizations \(r=\alpha_1\cdots\alpha_m=\beta_1\cdots\beta_n\). The elasticity of \(S\) is defined by \(\rho(S)=\sup_{r\in S}\rho(r)\). Let \(K\) be an algebraic number field, \(L/K\) its finite extension, \(Z_L\) the ring of integers of \(L\), and put \(S=N_{L/K}(Z_L)\). The author studies the relationship between \(\rho(Z_L\setminus\{0\})\) and \(\rho(S)\) and shows that if \(L/K\) is normal, then \(\rho(Z_L\setminus\{0\})\geq\rho(S)\). Examples are given to show that this may fail for non-normal extensions.