Simple module: a nonzero module with no proper nonzero submodule

Depends on

• Submodule of a module
• Unital left and right modules over a ring; unqualified module means left module

Used by

• Under the dictionary, subrepresentations are exactly submodules and irreducible representations are exactly simple modules Corollary
• A Deligne kernel need not be one external tensor factor Counterexample
• A perfect symmetrizing trace does not imply semisimplicity Counterexample
• A rank-one Hecke algebra with a perfect symmetrizing form and degenerate regular trace Counterexample
• Modular Specht modules need not be simple, and form heads can vanish Counterexample
• Composition series and length of a module Definition
• Primitive ideals of an enveloping algebra Definition
• Real, complex, and quaternion coefficient algebras and matrix notation Definition
• Semisimple modules as direct sums of simple modules Definition
• The radical, socle, head, and Loewy series of a finite-dimensional module Definition
• The socle as the sum of all simple submodules Definition
• A finite right exact functor needs no infinite-coproduct hypothesis Example
• A nonsplit simple has Hom-pairing diagonal two Example
• A tangent line and conic have one intersection point of local length two Example
• Decomposition matrices of S3 at p=2 and p=3 Example
• Explicitly lifting the two simple labels in a formal rank-two algebra Example
• GL₁ and the trivial parabolic endpoints Example
• The dual numbers are symmetric Frobenius and nonsemisimple over every field Example
• The dual numbers have Cartan map multiplication by two Example
• The dual-numbers tensor functor is right exact but not left exact Example
• A nondegenerate regular-trace pairing forces semisimplicity Lemma
• A zero-dimensional projective scheme has finitely many closed points with finite-dimensional local rings Lemma
• Dixmier's lemma: endomorphisms of a simple module over a countable-dimensional algebra Lemma
• Finite matrix and module preliminaries: right inverses, rank invariance, finite length and nilpotent trace Lemma
• Hilbert series and eventual Hilbert value of a two-form plane complete intersection Lemma
• Local flatness criterion by regular parameters Lemma
• Nonzero-form cellular heads, cell purity and shared-factor linkage Lemma
• A normal p-subgroup acts trivially on every simple module in characteristic p Proposition
• A primitive ideal determines a central character Proposition
• Finite-dimensional module categories satisfy the intrinsic finiteness conditions Proposition
• Primitive ideals are prime in the noncommutative sense Proposition
• Existence and uniqueness of cuspidal support Theorem
• For a finite-dimensional algebra, the Jacobson radical is nilpotent and the quotient by it is semisimple Theorem
• For a finite-dimensional algebra, the module radical is exactly the action of the Jacobson radical Theorem
• Indecomposable projective classes form a basis of split K0 Theorem
• Matrix rings over division rings are semisimple Theorem
• Nonzero cell heads classify the simple modules of a cellular algebra Theorem
• Projective and simple classes are dual bases under splitting Theorem
• Schur's lemma for simple modules Theorem
• Split formal deformations preserve simple-block labels; the trace determinant bounds exceptional parameters Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• McGerty, Algebra II: Rings and Modules, Section 3 (standard reference, not scraped)