Definition & Meaning | English word SUBMODULES

Definitions of SUBMODULES

1. plural of submodule.

Examples of Using SUBMODULES in a Sentence

• In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules.
• More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.
• The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct.
• A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.
• Schur's lemma says that a homomorphism between simple modules (modules with no non-trivial submodules) must be either zero or an isomorphism.
• In mathematics, specifically abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules.
• A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules.
• It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules.
• In addition to the fact rad(M) is the sum of superfluous submodules, in a Noetherian module rad(M) itself is a superfluous submodule.
• By contrast, unique decomposition into indecomposable submodules does not generalize as far, and the failure is measured by the ideal class group, which vanishes for PIDs.
• It is immediate that in a uniserial R-module M, all submodules except M and 0 are simultaneously essential and superfluous.
• A Demazure module is the b-submodule of V generated by the weight space of an extremal vector wλ, so the Demazure submodules of V are parametrized by the Weyl group W.
• To give a direct sum decomposition of a module into submodules is the same as to give orthogonal idempotents in the endomorphism ring of the module that sum up to the identity map.

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