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A heterocyclic compound or ring structure is a cyclic compound that has atoms of at least two different elements as members of its ring(s). [1] Heterocyclic organic chemistry is the branch of organic chemistry dealing with the synthesis, properties, and applications of organic heterocycles .
A homocycle or homocyclic ring is a ring in which all atoms are of the same chemical element. [1] A heterocycle or heterocyclic ring is a ring containing atoms of at least two different elements, i.e. a non-homocyclic ring. [2] A carbocycle or carbocyclic ring is a homocyclic ring in which all of the atoms are carbon. [3]
[citation needed] The term macrocycle is used when a ring-containing compound has a ring of 12 or more atoms. [6] [7] The term polycyclic is used when more than one ring appears in a single molecule. Naphthalene is formally a polycyclic compound, but is more specifically named as a bicyclic compound. Several examples of macrocyclic and ...
For example, if R is a commutative ring and f an element in R, then the localization [] consists of elements of the form /,, (to be precise, [] = [] / ().) [42] The localization is frequently applied to a commutative ring R with respect to the complement of a prime ideal (or a union of prime ideals) in R .
In bridged bicyclic compounds, the two rings share three or more atoms, separating the two bridgehead atoms by a bridge containing at least one atom. For example, norbornane, also known as bicyclo[2.2.1]heptane, can be viewed as a pair of cyclopentane rings each sharing three of their five carbon atoms. Camphor is a more elaborate example.
In mathematics, a Boolean ring R is a ring for which x 2 = x for all x in R, that is, a ring that consists of only idempotent elements. [1] [2] [3] An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive ...
Examples of limits and colimits in Ring include: The ring of integers Z is an initial object in Ring. The zero ring is a terminal object in Ring. The product in Ring is given by the direct product of rings. This is just the cartesian product of the underlying sets with addition and multiplication defined component-wise.
As an example, the nilradical of a ring, the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all n × n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. There are, however, analogues of the ...