This is an ideal because the sum of any two nilpotent elements is nilpotent ( by the binomial formula ), and the product of any element with a nilpotent element ( by commutativity ) is nilpotent.
22.
If the ring is a ring with no nilpotent elements, then the Frobenius endomorphism is injective : 0 } } means 0 } }, which by definition means that is nilpotent of order at most.
23.
By the Jacobson Morozov theorem every nilpotent element e \ in \ mathfrak { g } can be included into an \ mathfrak { sl _ 2 }-triple ( e, h, f ).
24.
Specifically, the set of all nilpotent elements in the ring of all " n " x " n " matrices over a division ring never forms an ideal, irrespective of the division ring chosen.
25.
If " k " is real, then the nilpotent elements are precisely those of finite additive order, and these in turn are the forms all of whose signatures are zero; " W " has Krull dimension 1.
26.
Finally, I do think there might be a problem in Semisimple algebra because there it is claimed that the radical is the unique nilpotent ideal containing all nilpotent elements of the algebra, and that finitedimensional algebras are semisimple iff the radical is trivial.
27.
The Springer resolution for Lie algebras is similar, except that " U " is replaced by the nilpotent elements of the Lie algebra of " G " and " X " replaced by the variety of Borel subalgebras.
28.
The same notations for the localization of a ring are used for modules : M _ \ mathfrak { p } denote the localization of " M " at a prime ideal \ mathfrak { p } and M _ f the localization of a non-nilpotent element " f ".
29.
A characteristic similar to that of Jacobson radical and annihilation of simple modules is available for nilradical : nilpotent elements of ring " R " are precisely those that annihilate all integral domains internal to the ring " R " ( that is, of the form " R " / " I " for prime ideals " I " ).
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