Module theory

generalization of vector space, with scalars in a ring instead of a field

in algebra, a module that has a basis

module over a ring with no nontrivial submodules

operation in abstract algebra

direct sum of irreducible modules

decomposition of an algebraic structure

module such that taking the tensor product with it induces an exact functor

in algebra, a module that is the direct summand of a free module

elements of a module space sent to 0 by regular elements of a ring

construction of a ring of fractions, in commutative algebra

algebraic construction

module that makes certain exact sequences split

integer invariant of a module in linear algebra

module which satisfies the descending chain condition on submodules

invariant of rings and modules in algebra

endomorphism algebra of an abelian group

In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in the sense that many of the relationships between left and right modules become simpler when they are expressed in terms of bimodules.

In algebra, a module that has a finite generating set

in algebra, an exact sequence which is used to describe the structure of an object

lemma stating that, given an ideal I in a Noetherian commutative ring R and a submodule N of a a finitely generated R-module M, there exists a positive integer k such that, for every n≥k, IⁿM ∩ N = Iⁿ⁻ᵏ(IᵏM ∩ N)

result in abstract algebra

concept in module theory

equivalence relation on rings

module without any strictly ascending sequence of submodules

finite-dimensional unital associative algebra with a compatible bilinear form

Prime ideal involved in a primary decomposition

concept in module theory

homological property of mathematical rings

concept in mathematics

operation that pairs a left and a right 𝑅‐module into an abelian group

linear representation of an associative algebra

annihilator of a simple module

module generated by one element

mathematical theorem

mathematical theorem