Primary factorization in semigroups
E. W. Johnson · Czechoslovak Mathematical Journal · 1986
Throughout, S will denote a commutative, multiplicative semigroup with 0 and 1. Factorization theory, in one form or another, has been a topic of ongoing interest in algebra since the beginnings of the subject.In this paper, we consider the implica tions of factorizations, of various types, of ideals as products of primary ideals.By a prime ideal, we shall mean an ideal P (ф5) which has the property that if it contains the product of two elements then it must contain one of them.The set M of all nonunits of S is a prime ideal, in fact the unique maximal ideal of S. By a pri mary ideal, we shall mean an ideal Q (ф5) which has the property that if it contains the product xy of two elements and fails to contain x, then it must contain a power of y.Any power of the maximal ideal M is easily seen to be primary.The radical of an ideal /, denoted rad(/), is the set of elements having a power in /.It is easy to see that an ideal P is prime iff whenever P contains the product of two ideals, it must contain one of them.Similarly, an ideal Q is primary iff whenever Q contains the product AB of two ideals and fails to contain A, rad(ô) must contain B, The radical of a primary ideal is prime, and any ideal having radical M is primary, as is easily seen.If Q is a primary ideal and rad(g) = P, then we will say that Q is P-primary or that Q is primary with associated prime P. We shall say that a semigroup has a primary decomposition theory if every ideal has a representation as a finite intersection of primary ideals (i.e., a primary decomposition).If S is Noetherian (i.e., satisfies A.C.C. on ideals) then every ideal has a primary decomposition.Any primary decomposition can be refined to a normal decomposition (i.e., one which is as short as possible and in which distinct primary terms have distinct radicals).If S is Noetherian then every primary ideal contains a power of its associated prime.We shall say thsit S has a strong primary decomposition theory if S has a primary decomposition theory and every primary contains a power of its radical.By an irreducible ideal, we shall mean a nonzero ideal which cannot be properly factored (i.e., A = ВС impUes В = S or С = S).If A and В are subsets of S then we shall use A : В to denote the set of all elements x such that xB a A. If A is an ideal of S, then Л : P is an ideal of S. If x is any element of S and A is an ideal of S, then A n (x) = (A: (x)) (x), as is easily seen.Hence a principal ideal is a factor of any ideal which it contains.By a principally reduced semigroup we shall mean a semi-