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A ring R is called a Jacobson ring if the nilradical and Jacobson radical of R/P coincide for all prime ideals P of R. An Artinian ring is Jacobson, and its nilradical is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is Noetherian), then it is nilpotent.
The ICD-10 Clinical Modification (ICD-10-CM) is a set of diagnosis codes used in the United States of America. [1] It was developed by a component of the U.S. Department of Health and Human services, [ 2 ] as an adaption of the ICD-10 with authorization from the World Health Organization .
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 ...
In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical n i l ( g ) {\displaystyle {\mathfrak {nil}}({\mathfrak {g}})} of a finite-dimensional Lie algebra g {\displaystyle {\mathfrak {g}}} is its maximal nilpotent ideal , which exists because the sum of any two nilpotent ideals is nilpotent.
The Baer radical of a ring is the intersection of the prime ideals of the ring R. Equivalently it is the smallest semiprime ideal in R. The Baer radical is the lower radical of the class of nilpotent rings. Also called the "lower nilradical" (and denoted Nil ∗ R), the "prime radical", and the "Baer-McCoy
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.
ICD-10 is the 10th revision of the International Classification of Diseases (ICD), a medical classification list by the World Health Organization (WHO). It contains codes for diseases, signs and symptoms, abnormal findings, complaints, social circumstances, and external causes of injury or diseases. [1]
Nakayama's lemma states that if a finitely generated module M is equal to IM where I is the Jacobson radical, then M is zero. neat Occasionally used to mean "unramified". nilpotent Some power is zero. Can be applied to elements of a ring or ideals of a ring. See nilpotent. nilradical The nilradical of a ring is the ideal of nilpotent elements ...