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In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. [1] Algebraic structures include groups , rings , fields , modules , vector spaces , lattices , and algebras over a field .
Band diagram for Schottky barrier at equilibrium Band diagram for semiconductor heterojunction at equilibrium. In solid-state physics of semiconductors, a band diagram is a diagram plotting various key electron energy levels (Fermi level and nearby energy band edges) as a function of some spatial dimension, which is often denoted x. [1]
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods ...
Diagram of the 26 sporadic simple groups, showing subquotient relationships. The monster contains 20 of the 26 sporadic groups as subquotients. This diagram, based on one in the book Symmetry and the Monster by Mark Ronan, shows how they fit together. [19] The lines signify inclusion, as a subquotient, of the lower group by the upper one.
In many situations in abstract algebra, such as for abelian groups, vector spaces or modules, the cokernel of the homomorphism f : X → Y is the quotient of Y by the image of f. In topological settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure of the image before passing to the quotient.
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The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference ...
The matrix algebra of n × n matrices over C with * given by the conjugate transpose. Its generalization, the Hermitian adjoint in the algebra of bounded linear operators on a Hilbert space also defines a *-algebra. The polynomial ring R[x] over a commutative trivially-*-ring R is a *-algebra over R with P *(x) = P (−x).
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