For an atomic domain $R$, its elasticity $\rho \left(R\right)$ is defined by : $\rho \left(R\right)=sup\{m/n\left|u_1\cdots u_m=v_1\cdots v_n\right.$ for irreducible $u_j,v_i\in R\}$. We study the elasticity of one-dimensional noetherian domains by means of the more subtle invariants ${\mu }_m\left(R\right)$ defined by : ${\mu }_m\left(R\right)=sup\{n\left|u_1\cdots u_m=u_1\cdots v_n\right.$ for irreducible $u_j,v_i\in R\}$. As a main result we characterize all orders in algebraic number fields having finite elasticity. On the way, we obtain a series of results concerning the invariants ${\mu }_m$ and $\rho$ for monoids and integral domains which are of independent interest.

We establish several finiteness characterizations and equations for the cardinality and the length of the set of overrings of rings with nontrivial zero divisors and integrally closed in their total ring of fractions. Similar properties are also obtained for related extensions of commutative rings that are not necessarily integral domains. Numerical characterizations are obtained for rings with some finiteness conditions afterwards.

2000 Mathematics Subject Classification: 16N80, 16S70, 16D25, 13G05.We construct some new examples showing that Heyman and Roos construction of the essential closure in the class of associative rings can terminate at any finite or the first infinite ordinal.