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Invariant Ring
In algebra, the fixed-point subring R^f of an automorphism ''f'' of a ring ''R'' is the subring of the fixed points of ''f'', that is, :R^f = \. More generally, if ''G'' is a group acting on ''R'', then the subring of ''R'' :R^G = \ is called the fixed subring or, more traditionally, the ring of invariants under . If ''S'' is a set of automorphisms of ''R'', the elements of ''R'' that are fixed by the elements of ''S'' form the ring of invariants under the group generated by ''S''. In particular, the fixed-point subring of an automorphism ''f'' is the ring of invariants of the cyclic group generated by ''f''. In Galois theory, when ''R'' is a field and ''G'' is a group of field automorphisms, the fixed ring is a subfield called the fixed field of the automorphism group; see Fundamental theorem of Galois theory. Along with a module of covariants, the ring of invariants is a central object of study in invariant theory. Geometrically, the rings of invariants are the coordinate rings ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variable (mathematics), variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in mathematical education, education, to the study of algebraic structures such as group (mathematics), groups, ring (mathematics), rings, and field (mathematics), fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
