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In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras
g
{\displaystyle {\mathfrak {g}}}
whose only ideals are {0} and
g
{\displaystyle {\mathfrak {g}}}
itself. It is important to emphasize that a one-dimensional Lie algebra (which is necessarily abelian) is by definition not considered a simple Lie algebra, even though such an algebra certainly has no nontrivial...
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In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 (ε is nilpotent). The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε where a and b are uniquely determined real numbers. The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of n dimensions leads to the Grassmann numbers.
The algebra of dual numbers is a ring that is a local ring since the principal ideal...
