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A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness. Torsionless A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5 ...
2. The grade grade(M) of a module M over a ring R is grade(Ann M,R), which for a finitely generated module over a Noetherian ring is the smallest n such that Ext n R (M,R) is non-zero. 3. The grade of a module M over a Noetherian local ring with maximal ideal I is the grade of m on I. This is also called the depth of M. This is not consistent ...
However, every projective module is a free module if the ring is a principal ideal domain such as the integers, or a (multivariate) polynomial ring over a field (this is the Quillen–Suslin theorem). Projective modules were first introduced in 1956 in the influential book Homological Algebra by Henri Cartan and Samuel Eilenberg.
In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem [1] — governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely generated modules.
The converse holds because the decomposition of 2. is equivalent to a decomposition into minimal left ideals = simple left submodules. The equivalence 1. 3. holds because every module is a quotient of a free module, and a quotient of a semisimple module is semisimple.
For example, the expression "5 mod 2" evaluates to 1, because 5 divided by 2 has a quotient of 2 and a remainder of 1, while "9 mod 3" would evaluate to 0, because 9 divided by 3 has a quotient of 3 and a remainder of 0. Although typically performed with a and n both being integers, many computing systems now allow other types of numeric operands.
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.
Any C*-algebra is a Hilbert -module with the action given by right multiplication in and the inner product , =. By the C*-identity, the Hilbert module norm coincides with C*-norm on A {\displaystyle A} .