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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.
If M is a (right) comodule over the coalgebra C, then M is a (left) module over the dual algebra C ∗, but the converse is not true in general: a module over C ∗ is not necessarily a comodule over C. A rational comodule is a module over C ∗ which becomes a comodule over C in the natural way.
A right R-module M R is defined similarly in terms of an operation · : M × R → M. Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be ...
Let R 1, R 2, R 3, R be rings, not necessarily commutative. For an R 1-R 2-bimodule M 12 and a left R 2-module M 20, is a left R 1-module. For a right R 2-module M 02 and an R 2-R 3-bimodule M 23, is a right R 3-module.
In mathematics, many types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures may be viewed in different ways, however the common starting point of algebra texts is that an algebraic object incorporates one or more sets with one or more binary operations or unary operations satisfying a ...
If N is also a ring (and hence an R-algebra), then this is the presentation of the N-module ; that is, the presentation extends under base extension. For left-exact functors , there is for example Proposition — Let F , G be left-exact contravariant functors from the category of modules over a commutative ring R to abelian groups and θ a ...