-
condition on
submodules, that is,
every increasing chain of
submodules becomes stationary after finitely many steps. Equivalently,
every submodule is finitely...
- mathematics,
especially in the
field of
module theory, the
concept of pure
submodule provides a
generalization of
direct summand, a type of
particularly well-behaved...
- let S and T be
submodules of M. Then: The sum S + T = {s + t | s ∈ S, t ∈ T} is a
submodule of M, The
intersection S ∩ T is a
submodule of M, and The quotient...
- For an R-module A, a
maximal submodule M of A is a
submodule M ≠ A
satisfying the
property that for any
other submodule N, M ⊆ N ⊆ A
implies N = M or...
- M with a
submodule N, the
module M is said to be an
essential extension of N (or N is said to be an
essential submodule or
large submodule of M) if for...
- and
module theory, each
right (resp. left) R-module M has a
singular submodule consisting of
elements whose annihilators are
essential right (resp. left)...
- ring. The
torsion submodule of a
module is the
submodule formed by the
torsion elements (in
cases when this is
indeed a
submodule, such as when the ring...
-
module theory, a
dense submodule of a
module is a
refinement of the
notion of an
essential submodule. If N is a
dense submodule of M, it may alternatively...
-
stating that
every submodule of a
finitely generated module over a
Noetherian ring is a
finite intersection of
primary submodules. This
contains the case...
- also
belong to I {\displaystyle I} . (Equivalently, if it is a
graded submodule of R {\displaystyle R} ; see § Graded module.) The
intersection of...
