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The concept of the Jacobson radical of a ring; that is, the intersection of all right (left) annihilators of simple right (left) modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right (left) ideals in the ring, shows how the internal structure of the ring is reflected by ...
The group of units of a topological ring is a topological group when endowed with the topology coming from the embedding of into the product as (,). However, if the unit group is endowed with the subspace topology as a subspace of , it may not be a topological group, because inversion on need not be continuous with respect to the subspace topology.
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
The generalization of the Zariski topology to the set of prime ideals of a commutative ring follows from Hilbert's Nullstellensatz, that establishes a bijective correspondence between the points of an affine variety defined over an algebraically closed field and the maximal ideals of the ring of its regular functions. This suggests defining the ...
The prime spectrum of a Boolean ring (e.g., a power set ring) is a compact totally disconnected Hausdorff space (that is, a Stone space). [4] (M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a spectral space) if and only if it is compact, quasi-separated and sober. [5]
The Topologist's sine curve, a useful example in point-set topology.It is connected but not path-connected. In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.
The stalk at a point can be thought of as the set of all germs of continuous functions at ; this is a local ring with the unique maximal ideal consisting of those germs whose value at is . If X {\displaystyle X} is a manifold with some extra structure, we can also take the sheaf of differentiable , or holomorphic functions.
Realpolitik (/ r eɪ ˈ ɑː l p ɒ l ɪ ˌ t iː k / ray-AHL-po-lih-teek German: [ʁeˈaːlpoliˌtiːk] ⓘ; from German real 'realistic, practical, actual' and Politik 'politics') is the approach of conducting diplomatic or political policies based primarily on considerations of given circumstances and factors, rather than strictly following ideological, moral, or ethical premises.