A set S in which a multiplication ab is defined is called a semigroup if this multiplication is associative and commutative, if an identity element is present in S, and if the cancellation law holds: ab = ac implies b = c. For example, the non-zero elements of a domain of integrity constitute a semigroup under multiplication. The notions of divisibility and irreducible elements are defined in the usual way. A semigroup S will be called an arithmetic if every element of S is uniquely decomposable into irreducible elements. The problem is to give necessary and sufficient conditions that a semigroup must satisfy in order that it can be embedded in an arithmetic. Any arithmetic z containing a semigroup S will be called an ideal arithmetic of S. This is an exact formulation of the problem of "restoring unique decomposition by the adjunction of ideal elements." The statement of the problem is in no way altered if we are considering a domain of integrity D rather than a semigroup. In that case we endeavor to embed the multiplicative semigroup S of D in an arithmetic A, or rather the semigroup S of principal ideals in D, which may be regarded as arising from S by identifying elements which divide each other. In Dedekind's theory of algebraic numbers, 2 is the set of all integral ideals in D, and it should be observed that z is not again a ring but only a semigroup-we can multiply two ideals but we cannot add them! This observation should make it clear that the problem is actually one of multiplication alone.' The general solution of the problem has been effected by means of what Krull calls v-ideals.2 A subset a of a domain of integrity D is a v-ideal if it contains every element of D which is divisible by all common divisors of a in the quotient-field of D. (Every v-ideal is also a Dedekind ideal, but not in general conversely.) Since this definition involves only the notion of divisibility, it can be applied to a semigroup S; of course we replace "quotient-field of D" by "quotient-group of S," the group of all formal quotients of elements