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Advanced Placement (AP) United States Government and Politics (often shortened to AP Gov or AP GoPo and sometimes referred to as AP American Government or simply AP Government) is a college-level course and examination offered to high school students through the College Board's Advanced Placement Program.
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).
The Oxford English Dictionary traces usage of 'radical' in a political context to 1783. [2] The Encyclopædia Britannica records the first political usage of 'radical' as ascribed to Charles James Fox, a British Whig Party parliamentarian who in 1797 proposed a 'radical reform' of the electoral system to provide universal manhood suffrage, thereby idiomatically establishing the term 'Radicals ...
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.
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.
In popular usage, reactionary refers to a strong traditionalist conservative political perspective of a person opposed to social, political, and economic change. [ 3 ] [ 4 ] Reactionary ideologies can be radical in the sense of political extremism in service to re-establishing past conditions.
Advanced Placement (AP) examinations are exams offered in United States by the College Board and are taken each May by students. The tests are the culmination of year-long Advanced Placement (AP) courses, which are typically offered at the high school level. AP exams (with few exceptions [1]) have a multiple-choice section and a free-response ...
A / nil(A) is a semisimple ring, where nil(A) is the nilradical of A. [citation needed] Every finitely generated module over A has finite length. (see above) A has Krull dimension zero. [6] (In particular, the nilradical is the Jacobson radical since prime ideals are maximal.) is finite and discrete.