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The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal.This is an ideal because the sum of any two nilpotent elements is nilpotent (by the binomial formula), and the product of any element with a nilpotent element is nilpotent (by commutativity).
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent. [1] [2]The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil.
Transformation in economics refers to a long-term change in dominant economic activity in terms of prevailing relative engagement or employment of able individuals. Human economic systems undergo a number of deviations and departures from the "normal" state, trend or development.
If R is commutative, the Jacobson radical always contains the nilradical. If the ring R is a finitely generated Z-algebra, then the nilradical is equal to the Jacobson radical, and more generally: the radical of any ideal I will always be equal to the intersection of all the maximal ideals of R that contain I. This says that R is a Jacobson ring.
Economic transformation can be measured through production/value-added measures and trade-based measures. Production-based measures include: (1) sector value added and employment data, to show productivity gaps between sectors; and (2) firm-level productivity measures, to examine average productivity levels of firms within one sector.
Also called resource cost advantage. The ability of a party (whether an individual, firm, or country) to produce a greater quantity of a good, product, or service than competitors using the same amount of resources. absorption The total demand for all final marketed goods and services by all economic agents resident in an economy, regardless of the origin of the goods and services themselves ...
It is also contested how much these changes were the result of neoliberal economic policies and how much they were the result of other factors; [137] in particular, some scholars argue that after the Crisis of 1982 the "pure" neoliberalism of the late 1970s was replaced by a focus on fostering a social market economy that mixed neoliberal and ...
The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring R/I is reduced if and only if I is a radical ...