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"Token Ring is an example of a ring topology." 802.5 (Token Ring) networks do not use a ring topology at layer 1. Token Ring networks are technologies developed by IBM typically used in local area networks. Token Ring (802.5) networks imitate a ring at layer 2 but use a physical star at layer 1. "Rings prevent collisions." The term "ring" only ...
A network's logical topology is not necessarily the same as its physical topology. For example, the original twisted pair Ethernet using repeater hubs was a logical bus topology carried on a physical star topology. Token Ring is a logical ring topology, but is wired as a physical star from the media access unit.
First, there is the notion of constructible topology: given a ring A, the subsets of of the form (),: satisfy the axioms for closed sets in a topological space. This topology on Spec ( A ) {\displaystyle \operatorname {Spec} (A)} is called the constructible topology.
An example of this situation is the adele ring of a global field; its unit group, called the idele group, is not a topological group in the subspace topology. If inversion on R × {\displaystyle R^{\times }} is continuous in the subspace topology of R {\displaystyle R} then these two topologies on R × {\displaystyle R^{\times }} are the same.
Metro Ethernet uses a fibre optic ring as a Gigabit Ethernet MAN backbone within a larger city. The ring topology is implemented using Internet Protocol (IP) so that data can be rerouted if a link is congested or fails. [6] In the US the Sprint was an early adopter of fibre optic rings that routed IP packets on the MAN backbone.
In knot theory, the Borromean rings are a simple example of a Brunnian link, a link that cannot be separated but that falls apart into separate unknotted loops as soon as any one of its components is removed. There are infinitely many Brunnian links, and infinitely many three-curve Brunnian links, of which the Borromean rings are the simplest.
In commutative algebra, the filtration on a commutative ring R by the powers of a proper ideal I determines the Krull (after Wolfgang Krull) or I-adic topology on R.The case of a maximal ideal = is especially important, for example the distinguished maximal ideal of a valuation ring.
For example, n-by-n matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring. A λ-ring is a commutative ring R together with operations λ n : R → R that are like n th exterior powers :