The fundamental theorem of arithmetic applies to any Euclidean domain : Any number from a Euclidean domain can be factored uniquely into irreducible elements.
12.
The irreducible elements of sowei are a helmet form, lustrous blackness, beautifully arranged hair, downcast eyes, a closed mouth and an expression of inner spiritual concentration.
13.
The unique factorization property is a direct consequence of Euclid's lemma : If an irreducible element divides a product, then it divides one of the factors.
14.
There exist lattices in which the join-prime elements form a proper subset of the join-irreducible elements, but in a distributive lattice the two types of elements coincide.
15.
However, if D is a GCD domain, and x is an irreducible element of D, then the ideal generated by x " is " a prime ideal of D.
16.
Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in general that an ideal generated by an irreducible element is an irreducible ideal.
17.
The lattice ordering on the subset of join-irreducible elements forms a partial order; Birkhoff's theorem states that the lattice itself can be recovered from the lower sets of this partial order.
18.
The notion of irreducible element is defined in any integral domain : an element is irreducible if it is not invertible and cannot be written as a product of two non-invertible elements.
19.
The module is determined up to isomorphism by specifying its free rank, and for class of associated irreducible elements and each positive integer the number of times that occurs among the elementary divisors.
20.
The performativity of the event exceeds all anticipation or programming, and hence contains an irreducible element of messianism, linking it to justice ( here distinguished from law ), as well as to revolution.
