Original sentence: As the ring axioms are very general, often it is found that structurally, apart from the ring axioms themselves, rings have very little in common with the integers.For example, a ring theoretic analogue of the fundamental theorem of arithmetic exists, as well as a generalization of the Euclidean algorithm and Bézout's identity.However, this is not to say that these results hold in the context of all rings.In general, many properties of the integers, relevant to only the operations of addition and multiplication, can be generalized to the context of rings (notice that subtraction and division are inverses of the operations of addition and multiplication respectively, and therefore can also be defined in the context of rings (under certain conditions)).It is therefore necessary to impose additional conditions on the structure of rings, that allow results such as the fundamental theorem of arithmetic to hold true.Rings, being structurally similar to the integers in many ways, have algebraic properties similar to the integers.That is, apart from the ring axioms that are shared by both the ring, and the set of integers, there are not many other properties that both structures satisfy
Edited sentence: A ring retains enough structure from the integers, to allow the notion of a prime number to be well-defined.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, or UFD in short.It is the fundamental theorem of arithmetic, that every integer greater than one, is the unique product of positive prime numbers, or more generally, any integer is the product of prime numbers, either positive or negative.Likewise, rings which support a translation of the Euclidean algorithm or Bézout's identity, are known as Euclidean domains, and Bézout domains, respectively.Rings share various structural aspects in common with the integers, and in particular, number-theoretic properties.Therefore, although all rings share many structural aspects of the integers in common, not all structural aspects of the integers are retained