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An ideal in a ring is radical if and only if the quotient ring is reduced. The radical of a homogeneous ideal is homogeneous. The radical of an intersection of ideals is equal to the intersection of their radicals: . The radical of a primary ideal is prime. If the radical of an ideal is maximal, then is primary.
By a slight change in notation (and viewpoint), for an interval [ a, b ], the difference quotient. is called [5] the mean (or average) value of the derivative of f over the interval [ a, b ]. This name is justified by the mean value theorem, which states that for a differentiable function f, its derivative f′ reaches its mean value at some ...
The quotient module R/A is a simple right R-module. Maximal right/left/two-sided ideals are the dual notion to that of minimal ideals. Examples. If F is a field, then the only maximal ideal is {0}. In the ring Z of integers, the maximal ideals are the principal ideals generated by a prime number.
Ideal quotient. In abstract algebra, if I and J are ideals of a commutative ring R, their ideal quotient ( I : J) is the set. Then ( I : J) is itself an ideal in R. The ideal quotient is viewed as a quotient because if and only if . The ideal quotient is useful for calculating primary decompositions. It also arises in the description of the set ...
Numerical differentiation. Finite difference estimation of derivative. In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function.
An ideal can be used to construct a quotient ring in a way similar to how, in group theory, a normal subgroup can be used to construct a quotient group. Among the integers, the ideals correspond one-for-one with the non-negative integers: in this ring, every ideal is a principal ideal consisting of the multiples of a single non-negative number ...
In mathematics, taking the nth root is an operation involving two numbers, the radicand and the index or degree. Taking the nth root is written as , where x is the radicand and n is the index (also sometimes called the degree). This is pronounced as "the nth root of x". The definition then of an nth root of a number x is a number r (the root ...
Rationalisation (mathematics) In elementary algebra, root rationalisation is a process by which radicals in the denominator of an algebraic fraction are eliminated. If the denominator is a monomial in some radical, say with k < n, rationalisation consists of multiplying the numerator and the denominator by and replacing by x (this is allowed ...