Original sentence: However, ring theory studies rings in their full generality, and often the ring axioms are all that is necessary to develop a rich theory.However, the conclusion of the fundamental theorem of arithmetic is not satisfied by all rings - those rings which do allow a translation of this theorem to the context of ring theory, are known as unique factorization domains.Likewise, rings which support a translation of the Euclidean algorithm or Bézout's identity, are known as Euclidean domains, and Bézout domains, respectively.Therefore, although all rings share many structural aspects of the integers in common, not all structural aspects of the integers are retained
Edited sentence: However, the conclusion of the fundamental theorem of arithmetic is not satisfied by all rings - those rings which do support a translation of this theorem into the context of ring theory, are known as unique factorization domains.Likewise, rings which support an analogue of the Euclidean algorithm or Bézout's identity, are known as Euclidean domains, and Bézout domains, respectively.That said, it is often the case that the ring axioms are sufficient to develop a vast and extensive theory, not necessarily at all similar to that of the integers.Therefore, although every ring shares many number-theoretic and algebraic properties in common with the integers, the similarity ends there for many rings