Emmy Noether concerned the theory of rings. In particular Noether was able to develop a decomposition theory for ideals that was applicable to rings. This was more general than the rings of integers in algebraic number fields. These rings are called Noetherian Rings, communative rings with identity which satisfy the ascending chain condition, that every chain of ideals in the ring such that breaks off after a finite number of terms.
R has a multiplicative identity element.
R is an integral domain, meaning R has no zero divisors. R is integrally closed in its qoutient field. In other words, every element of the field which satisfies a polynomial equation over the ring is itself in the ring . She then went on to show that if R satisfied these five axioms, then its integral closure in a finite separable extension of its qoutient field also satisfies them. Writing over 44 papers, Emmy made numerous contributions to algebraic number theory all of which cannot be discussed here.