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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).
A conservative replacement (also called a conservative mutation or a conservative substitution or a homologous replacement) is an amino acid replacement in a protein that changes a given amino acid to a different amino acid with similar biochemical properties (e.g. charge, hydrophobicity and size).
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 ...
The free radical theory of aging states that organisms age because cells accumulate free radical damage over time. [1] A free radical is any atom or molecule that has a single unpaired electron in an outer shell. [2] While a few free radicals such as melanin are not chemically reactive, most biologically relevant free radicals are highly ...
The mitochondrial theory of ageing has two varieties: free radical and non-free radical. The first is one of the variants of the free radical theory of ageing. It was formulated by J. Miquel and colleagues in 1980 [1] and was developed in the works of Linnane and coworkers (1989). [2] The second was proposed by A. N. Lobachev in 1978. [3]
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 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.
This glossary of biology terms is a list of definitions of fundamental terms and concepts used in biology, the study of life and of living organisms.It is intended as introductory material for novices; for more specific and technical definitions from sub-disciplines and related fields, see Glossary of cell biology, Glossary of genetics, Glossary of evolutionary biology, Glossary of ecology ...