Such an editor has no access to the more obvious interpretants of " feeling ", and may thus help salvage even for the native reader the irreducible element of " knowledge " in his culture's songs.
22.
Another equivalent characterization of finite join-distributive lattices is that they are graded ( any two maximal chains have the same length ), and the length of a maximal chain equals the number of meet-irreducible elements of the lattice.
23.
The elementary divisors can be obtained from the list of invariant factors of the module by decomposing each of them as far as possible into pairwise relatively prime ( non-unit ) factors, which will be powers of irreducible elements.
24.
Note that in a PID, the nonzero primary ideals are powers of prime ideals, so the elementary divisors can be written as powers q _ i = p _ i ^ { r _ i } of irreducible elements.
25.
For non-atoroidal automorphisms, Bestvina and Handel produce an example of an irreducible but not fully irreducible element of Out ( F _ n ), induced by a suitably chosen pseudo-Anosov homeomorphism of a surface with more than one boundary component.
26.
It is a discrete valuation ring; the " unique " irreducible element is " X " and the valuation assigns to each function " f " the order ( possibly 0 ) of the zero of " f " at 0.
27.
Note that an irreducible element of " R " is still irreducible when viewed as constant polynomial in " R " [ " X " ]; this explains the need for " non-constant " above, and in the irreducibility statements below.
28.
If in addition is Gauss's lemma is irreducible in, whether or not it is primitive ( since constant factors are invertible in ); in this case a possible choice of prime ideal is the principal ideal generated by any irreducible element of.
29.
Since the square root of an algebraic integer is again an algebraic integer, it is not possible to factor any nonzero nonunit algebraic integer into a finite product of irreducible elements, which implies that \ overline { \ textbf { Z } } is not Noetherian!
30.
For each irreducible element such that some power occurs in, take the highest such power, removing it from, and multiply these powers together for all ( classes of associated ) to give the final invariant factor; as long as is non-empty, repeat to find the invariant factors before it.
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