Math 711, Fall 2007 Problem Set #5 Solutions 1. (a) The extension
... 5(b) in Problem Set #3 gives x(z 2 )q ∈ (xq , y q ) for all q.) Hence, no parameter ideal is tightly closed in R. Since z 2 is a socle generator mod (x, y), (xy)z 2 is mod (x2 , y 2 ). Mod J = (x2 , y 2 , xyz 2 ) the ring is K-spanned by 1, x, y, z, xy, xz, yz, z 2 , xz 2 , yz 2 and xyz. It suffices ...
... 5(b) in Problem Set #3 gives x(z 2 )q ∈ (xq , y q ) for all q.) Hence, no parameter ideal is tightly closed in R. Since z 2 is a socle generator mod (x, y), (xy)z 2 is mod (x2 , y 2 ). Mod J = (x2 , y 2 , xyz 2 ) the ring is K-spanned by 1, x, y, z, xy, xz, yz, z 2 , xz 2 , yz 2 and xyz. It suffices ...
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly n nth roots of unity, if n is not divisible by the characteristic of the field.