Algebra Notes

ideal

For an arbitrary ring ( R , + , ⋅ ), let (R,+) be its additive group.

A subset I is called a two-sided ideal (or simply an ideal) of R if it is an additive subgroup of R that “absorbs multiplication by elements of R.”

i.e.

• (I,+) is a subgroup of ( R , + )
• ∀ x ∈ I , ∀ r ∈ R : x ⋅ r , r ⋅ x ∈ I

homogeneous polynomial

In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree. For example,

x^5 + 2 x^3 y^2 + 9 x y^4

is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.

A polynomial is homogeneous if and only if it defines a homogeneous function.

homogeneous function

a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

For example, a homogeneous function of two variables x and y is a real-valued function that satisfies the condition f ( α x , α y ) = α k f ( x , y )