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Tag Archives: simple modules
Commutative Algebra 37
Artinian Modules Instead of the ascending chain condition, we can take its reverse. Definition. Let M be an A-module. Consider the set of submodules of M, ordered by inclusion, i.e. if and only if . We say M is artinian … Continue reading
Posted in Advanced Algebra
Tagged artinian, composition factors, composition series, length of module, noetherian, simple modules
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Modular Representation Theory (III)
Let’s work out some explicit examples of modular characters. First, we have a summary of the main results. Let be the modular characters of the simple k[G]-modules; they form a basis of Let be those of the projective indecomposable k[G]-modules; they form a basis … Continue reading
Noetherian and Artinian Rings and Modules
We saw the case of the semisimple ring R, which is a (direct) sum of its simple left ideals. Such a ring turned out to be nothing more than a finite product of matrix algebras. One asks if there is a … Continue reading
Posted in Notes
Tagged algebra, artinian, noetherian, semisimple rings, simple modules
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The Group Algebra (III)
As alluded to at the end of the previous article, we shall consider the case where K is algebraically closed, i.e. every polynomial with coefficients in K factors as a product of linear polynomials. E.g. K = C is a common choice. Having assumed … Continue reading
Posted in Notes
Tagged character theory, division rings, group algebras, quaternions, semisimple rings, simple modules
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The Group Algebra (I)
[ Note: the contents of this article overlap with a previous series on character theory. ] Let K be a field and G a finite group. The group algebra K[G] is defined to be a vector space over K with basis , where “g” here is … Continue reading
Posted in Notes
Tagged character theory, group actions, group algebras, modules, representation theory, semisimple rings, simple modules
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Simple Modules
We briefly talked about modules over a (possibly non-commutative) ring R. An important aspect of modules is that unlike vector spaces, modules are usually not free, i.e. they don’t have a basis. For example, take the Z-module given by Z/2Z. [ Recall: a Z-module is … Continue reading
Posted in Notes
Tagged algebra, associative algebra, division rings, rings, schur's lemma, simple modules
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Elementary Module Theory (I)
Modules can be likened to “vector spaces for rings”. To be specific, we shall see later that a vector space is precisely a module over a field (or in some cases, a division ring). This set of notes assumes the … Continue reading
Posted in Notes
Tagged generated submodules, ideals, left ideals, modules, rings, scalar multiplication, simple modules, submodules
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Quick Guide to Character Theory (II): Main Theory
Reminder: throughout this series, G is a finite group and K is a field. All K-vector spaces are assumed to be finite-dimensional over K. G4. Maschke’s Theorem If is a K[G]-submodule, it turns out V is isomorphic to the direct sum of W and some other submodule W’. … Continue reading
Mathematics and Such 