This is not true for irreducible ideals : an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in \ mathbb Z for the ideal 4 \ mathbb Z : It is not the intersection of two strictly greater ideals.
32.
The " unique " irreducible element is " T ", the maximal ideal of " R " is the principal ideal generated by " T ", and the valuation ? assigns to each power series the index ( i . e . degree ) of the first non-zero coefficient.
33.
At a fund-raiser for moderate Republican candidates held in New York City last spring, several leading Republicans pleaded for tolerance within the party for differing views on abortion rights, but Mrs . Whitman rejected a soothing compromise on the issue, insisting in her speech that respect for abortion rights should be an irreducible element of Republican Party philosophy.
34.
The condition that " R " is a GCD domain is not superfluous because it implies that every irreducible element of this ring is also a prime element, which in turn implies that every nonzero element of " R " has at most one factorization into a product of irreducible elements and a unit up to order and associate relationship.
35.
The condition that " R " is a GCD domain is not superfluous because it implies that every irreducible element of this ring is also a prime element, which in turn implies that every nonzero element of " R " has at most one factorization into a product of irreducible elements and a unit up to order and associate relationship.
36.
In this way the concept'empty space'loses its meaning . . . . The field thus becomes an irreducible element of physical description, irreducible in the same sense as the concept of matter ( particles ) in the theory of Newton . . . . The physical reality of space is represented by a field whose components are continuous functions of four independent variables-the co-ordinates of space and time.
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