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- Quotient_module abstract "In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic. It is the same construction used for quotient groups and quotient rings.Given a module A over a ring R, and a submodule B of A, the quotient space A/B is defined by the equivalence relation a ~ b if and only if b − a is in B,for any a and b in A. The elements of A/B are the equivalence classes [a] = { a + b : b in B }.The addition operation on A/B is defined for two equivalence classes as the equivalence class of the sum of two representatives from these classes; and in the same way for multiplication by elements of R. In this way A/B becomes itself a module over R, called the quotient module. In symbols, [a] + [b] = [a+b], and r·[a] = [r·a], for all a,b in A and r in R.".
- Quotient_module wikiPageID "364820".
- Quotient_module wikiPageLength "2598".
- Quotient_module wikiPageOutDegree "18".
- Quotient_module wikiPageRevisionID "613453739".
- Quotient_module wikiPageWikiLink Abstract_algebra.
- Quotient_module wikiPageWikiLink Addition.
- Quotient_module wikiPageWikiLink Category:Module_theory.
- Quotient_module wikiPageWikiLink Complex_number.
- Quotient_module wikiPageWikiLink Equivalence_relation.
- Quotient_module wikiPageWikiLink If_and_only_if.
- Quotient_module wikiPageWikiLink Integer.
- Quotient_module wikiPageWikiLink Isomorphism.
- Quotient_module wikiPageWikiLink Modular_arithmetic.
- Quotient_module wikiPageWikiLink Module_(mathematics).
- Quotient_module wikiPageWikiLink Polynomial_ring.
- Quotient_module wikiPageWikiLink Quotient_group.
- Quotient_module wikiPageWikiLink Quotient_ring.
- Quotient_module wikiPageWikiLink Quotient_space_(topology).
- Quotient_module wikiPageWikiLink Real_number.
- Quotient_module wikiPageWikiLink Ring_(mathematics).
- Quotient_module wikiPageWikiLinkText "Quotient module".
- Quotient_module wikiPageWikiLinkText "quotient map".
- Quotient_module wikiPageWikiLinkText "quotient module".
- Quotient_module wikiPageWikiLinkText "quotient".
- Quotient_module wikiPageUsesTemplate Template:Reflist.
- Quotient_module subject Category:Module_theory.
- Quotient_module comment "In abstract algebra, given a module and a submodule, one can construct their quotient module. This construction, described below, is analogous to how one obtains the ring of integers modulo an integer n, see modular arithmetic. It is the same construction used for quotient groups and quotient rings.Given a module A over a ring R, and a submodule B of A, the quotient space A/B is defined by the equivalence relation a ~ b if and only if b − a is in B,for any a and b in A.".
- Quotient_module label "Quotient module".
- Quotient_module sameAs Q1432554.
- Quotient_module sameAs Quotientenmodul.
- Quotient_module sameAs Rilata_modulo.
- Quotient_module sameAs Module_quotient.
- Quotient_module sameAs 剰余加群.
- Quotient_module sameAs Moduł_ilorazowy.
- Quotient_module sameAs m.01_vyl.
- Quotient_module sameAs Q1432554.
- Quotient_module wasDerivedFrom Quotient_module?oldid=613453739.
- Quotient_module isPrimaryTopicOf Quotient_module.