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For example, the process of rewriting a fraction into one with the smallest whole-number denominator possible (while keeping the numerator a whole number) is called "reducing a fraction". Rewriting a radical (or "root") expression with the smallest possible whole number under the radical symbol is called "reducing a radical".
The oxidized and reduced forms are in fast equilibrium with the semiquinone form, shifted against the formation of the radical: [2] Fl ox + Fl red H 2 ⇌ FlH • where Fl ox is the oxidized flavin, Fl red H 2 the reduced flavin (upon addition of two hydrogen atoms) and FlH • the semiquinone form (addition of one hydrogen atom).
The most basic example of a semisimple module is a module over a field, i.e., a vector space. On the other hand, the ring Z of integers is not a semisimple module over itself, since the submodule 2Z is not a direct summand. Semisimple is stronger than completely decomposable, which is a direct sum of indecomposable submodules.
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 ...
Radical elimination can be viewed as the reverse of radical addition. In radical elimination, an unstable radical compound breaks down into a spin-paired molecule and a new radical compound. Shown below is an example of a radical elimination reaction, where a benzoyloxy radical breaks down into a phenyl radical and a carbon dioxide molecule. [7]
In commutative algebra, semiprime ideals are also called radical ideals and semiprime rings are the same as reduced rings. For example, in the ring of integers , the semiprime ideals are the zero ideal, along with those ideals of the form n Z {\displaystyle n\mathbb {Z} } where n is a square-free integer .
Initial electron transfer and loss of halide generate an organic radical, which may combine with a second molecule of samarium(II) iodide to form an organosamarium species. Protonation of this species then yields the reduced product. Alternatively, the intermediate organic radical may abstract a hydrogen atom from the solvent S–H.
For example, SL(n,Z) is an arithmetic subgroup of SL(n,Q). For a Lie group G, a lattice in G means a discrete subgroup Γ of G such that the manifold G/Γ has finite volume (with respect to a G-invariant measure). For example, a discrete subgroup Γ is a lattice if G/Γ is compact.