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