Abstract
This chapter discusses factor decompositions and divisibility. It presents a theorem which states that in a zero-divisor-free ring R with unity element 1 the units are identical with the left divisors of 1 and they form a subgroup with the same unity element 1, consisting of the invertible elements of R; further, a product of elements of R is a unit only if all the factors are units. In an integral domain R with irreducible factor decomposition this is unique if and only if all the irreducible elements in R are prime. In a ring with prime decomposition, irreducible element and prime element are equivalent terms. For historical reasons, the first term is more common in many such rings, while in others the second is more common.
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